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MCQ QUESTIONS APPLICATIONS OF DEFINITE INTEGRAL
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7) APPLICATIONS OF DEFINITE INTEGRAL
Application of Definite Integrals1. The area bounded by the line y=mx,X-axis and the ordinates x=a, x=b,(a<b) is
a)m (b2−a2)
3sq .units b)
m (a2−b2)3
sq .units
c)m (b2−a2)
2sq .units d)
m (a2−b2)2
sq .units
2. The area bounded by the line y=2x,X-axis and the ordinates x=1, x=2isa) 1 Sq.units b) 3 Sq.unitsc) 2 Sq.units d) 4 Sq.units
3. The area bounded by the line y=3x,X-axis and the ordinates x=1, x=3isa) 2 Sq.units b) 8 Sq.unitsc) 4 Sq.units d) 12 Sq.units
4. The area bounded by the line y=x,X-axis and the ordinates x=-1, x=2is [Rajastan P.E.T.2001]
a)52Sq .units b)
13Sq .units
c)32Sq .units d)
23Sq .units
5. The area bounded by the line y=mx+c,X-axis and the ordinates x=a, x=b,(a<b) isa) m (b2−a2)
2+c (b−a)sq .units
b) m (b2−a2)2
−c (b−a)sq .units
c) m (b2−a2)2
−c (a−b)sq .units
d) m (b2−a2)2
+c (a−b)sq .units
6. The area bounded by the line y=2x+3,X-axis and the ordinates x=2, x=5isa) 12 Sq.units b) 15 Sq.unitsc) 24 Sq.units d) 30 Sq.units
7. The area bounded by the line 2y=5x+7,X-axis and the ordinates x=2, x=8isa) 48 Sq.units b) 64 Sq.unitsc) 96 Sq.units d) 192 Sq.units
8. The area bounded by the line 2y=x-2,X-axis and the ordinates x=-4, x=-1isa)34Sq .units b)
94Sq .units
c)274Sq .units d)
814Sq .units
9. The area bounded by the curvey=x3,X-axis and the ordinates x=1, x=3isa) 5 Sq.units b) 10 Sq.unitsc) 20 Sq.units d) 40 Sq.unts
10. The area bounded by the curvey= 4x2
,X-axis and the ordinates x=1, x=3is
a)13Sq .units b)
23Sq .units
c)43Sq .units d)
83Sq .units
11. The area bounded by the curvey=2√1−x2,X-axis and the ordinates x=0, x=1isa)π2sq .units b)
3π2sq .units
c) π sq .units d) 3 π sq .units12. The area bounded by the curvey2(x2+6 x−55)=1X-axis and the ordinates x=7, x=14is
a) log2 sq.units b) 2log2 sq.unitsc) 3log2 sq.units d) 4log2 sq.units
13. Theareabounded by the loopof the curvey2=x2 (1−x ) is
a)25sq .unts b) 85sq.unts
c)215
sq .unts d)815
sq .unts
14. The area bounded by the curve xy=2,X-axis and the ordinates x=1, x=4isa) log 2 sq.units b) 2log 2 sq.unitsc) 3log 2 sq.units d) 4log 2 sq.units
15. The area bounded by the curve xy=4,X-axis and the ordinates x=1, x=3isa) log 3sq.units b) 2log 3sq.unitsc) 3log 3sq.units d) 4log 3sq.units
16. The area bounded by the curve xy=16,X-axis and the ordinates x=4, x=8isa) log 2 sq.units b) 2log 2 sq.unitsc) 4log 2 sq.units d) 16log 2 sq.units
17. The area bounded by the curve 2xy=1,X-axis and the ordinates x=2, x=8isa) log 2 sq.units b) 2log 2 sq.units
P a g e |1
c) 3log 2 sq.units d) 4log 2 sq.units18. The area bounded by the curvey=sin x,X-axis and the ordinates x=1, x=2 πis
a) 2 sq.units b) 4 sq.unitsc) 8 sq.units d) 16 sq.units
19. The area bounded by the curvey=sin2 x,X-axis and the ordinates x=π4 , x=3π4 is
a) 1 sq.units b) 2sq.unitsc)12sq .units d)
14sq .units
20. The area bounded by the curvey=cos x,X-axis and the ordinates x=0 tox=π2 is
a) 1 Sq.units b) 2 Sq.unitsc) 4 Sq.units d) 8 Sq.units
21. The area bounded by the curvey=cos x,X-axis and the ordinates x=0, x=πisa) 1 sq.units b) 2 sq.unitsc) 4 sq.units d) 8 sq.units
22. The area bounded by the curvey=cos x,X-axis and the ordinates x=0 tox=2 πisa) 2 sq.units b) 4 sq.unitsc) 8 sq.units d) 16 sq.units
23. The area bounded by the curvey=2cos x,X-axis and the ordinates x=0 to2π isa) 2 sq.units b) 4 sq.unitsc) 8 sq.units d) 16 sq.units
24. The area bounded by the curvey=cos3 x,0<x<π6 is
a)13sq .units b)
23sq .units
c)16sq .units d)
32sq .units
25. The area bounded by the curvey=sin2( x2 ),X-axis and the ordinates x=0 ,π6 isa) 2𝜋−2sq. units b) 4 (π−2 ) sq .units
c)12
(π−2 ) sq .units d)14
(π−2 ) sq .units
26. The area bounded by the curvey=2x+sin x,y=0 from x=0, x=π2 is
a) 1− π2
4sq .units b) 1+ π
2
4sq .units
c) 1− π2
2sq .units d) 1+ π
2
2sq .units
27. The area bounded by the curve y2=4ax and its latus rectum is [Rajastan P.E.T.1997,2000,2002]
a)4 a2
3sq .units b)
8a2
3sq .units
c)4 a√a3
sq .units d)8a√a3
sq .units
28. The area bounded by the curve y2=8 x and theline x=2 isa)43sq .units b)
83sq .units
c)163sq .units d)
323sq .units
29. The area bounded by the curve y2=16 x and theline x=4 isa)1283
sq .units b)643sq .units
c)323sq .units d)
163sq .units
30. The area bounded by the curve y2=4 x and theline x=1 ,x=4 above X-axis isa)73sq .units b)
143sq .units
c)283sq .units d)
563sq .units
31. The area bounded by the curve y2=4 x and theline x=2 ,x=4 above X-axis isa)43
(4−√2 ) sq .units b)83
(4−√2 ) sq .units
c)43
(2−√2 ) sq .units d)83
(2−√2 ) sq .units
32. The area bounded by the curve y2=9 x and theline x=4,x=9 above X-axis isa) 9 sq.units b) 18 sq.unitsc) 19 sq.units d) 38 sq.units
33. The area bounded by the curve x2=ayX-axis and the lines x=a isa)a2sq . units b)
a3sq .units
c) a2
2sq .units d) a
2
3sq .units
34. The area bounded by the curve x2=ay , X-axis and the linesx=a,x=2a isP a g e |2
a) 7a2
3sq .units b) 5a
2
3sq .units
c)7a3sq .units d)
5a3sq .units
35. The area bounded by the parabola x2=4 y , the lines y=2,y=4 and Y-axis in the first quadrant isa)43
(8−2√2 ) sq .units b)83
(8−2√2 ) sq .units
c)43
(4−2√2 ) sq .units d)83
(4−2√2 ) sq .units
36. The area bounded by the parabola x2=16 y , the lines y=1,y=4 and Y-axis in the first quadrant is
a)1123
sq .units b)563sq .units
c)283sq .units d)
143sq .units
37. The area bounded by the parabola y=4 x2 , x ≥0,Y-axis & the lines the lines y=1,y=4 is
a)13sq .units b)
23sq .units
c)53sq .units d)
73sq .units
38. The area bounded by the parabola y=√6 x+4,X-axis from x=0 to x=2 in the first quadrant is
a)569sq .units b)
289sq .units
c)569sq .units d)
283sq .units
39. The area bounded by the parabola y=4√ x−1 ,1≤ x≤3and X-axis in the first quadrant is
a)2√23
sq .units b)4 √23
sq .units
c)16√23
sq .units d)32√23
sq .units
40. The area bounded by the parabola y=2x−x2,X-axis is
a)23sq .units b)
32sq .units
c)43sq .units d)
42sq .units
41. The area bounded by the parabola y=4 x−x2, X-axis is [M.P.C.E.T.1999,2003]
a)43sq .units b)
83sq .units
c)163sq .units d)
323sq .units
42. The area bounded by the parabolay=16−x2 ,0≤ x≤4 and the co-ordinate axes is
a)163sq .units b)
323sq .units
c)643sq .units d)
1283
sq .units
43. The area bounded by the parabolay=4−x2X-axis and lines x=0,x=2 is
a)163sq .units b)
83sq .units
c)43sq .units d)
23sq .units
44. The area bounded by the parabolay=x2+3X-axis and lines x=0,x=3 is
a) 3 sq.units b) 6 sq.unitsc) 12 sq.units d) 18 sq.units
45. The area bounded by the parabolay=x2+1X-axis and lines x=0,x=3 is
a) 2 sq.units b) 3 sq.unitsc) 6 sq.units d) 12 sq.units
46. The area bounded by the parabolay=2−x2X-axis and lines x=-1,x=1 is
a)23sq .units b)
53sq .units
c)103sq .units d)
203sq .units
47. The area bounded by the parabolay=x2−3 xthe lines y=2x and X-axis is
a)73sq .units b)
143sq .units
c)283sq .units d)
493sq .units
48. The area bounded by the parabolay=x2−3 xthe lines y=2x is
a)253sq .units b)
1253
sq .units
c)256sq .units d)
1256
sq .units
P a g e |3
49. The area bounded by the curve x=a t 2 , y=2atBetween the ordinates corresponding to t=1 and t=2 is
a) 7a2
3sq .units b) 14a
2
3sq .units
c) 28a2
3sq .units d) 56a
2
3sq .units
50. The area bounded by the curve y=ex,x=0,x=2 about X-axis isa) e−1 sq .units b) e−2 sq .unitsc) e2−1 sq .units d) e2−2 sq .units
51. The area bounded by the curve y=log x, X-axis and the ordinates x=1 and x=e isa) e sq .units b) 2e sq .unitsc) log e sq .units d) 2 log e sq .units
52. The area enclosed by the circle of radius r is a) πr sq .units b) 2πr sq .unitsc) π r2 sq .units d) 2π r2 sq .units
53. The area enclosed by the circle x2+ y2=9 isa) 2π sq .units b) 4 π sq . unitsc) 3 π sq .units d) 9 π sq .units
54. The area enclosed by the circle x2+ y2=16 isa) 2π sq .units b) 4 π sq . unitsc) 8 π sq.units d) 16 π sq .units
55. The area enclosed by the circle x2+ y2=2ax from x=0 to x=a above X-axis is
a) π a2
2sq .units b) π a
2
4sq .units
c) π a2
3sq .units d) π a
2
6sq .units
56. The area bounded by the ellipse b2 x2+a2 y2=a2b2 is [Karnataka C.E.T.1993]
a) πabsq .units b) 2πab sq .unitsc)π2ab sq .units d)
3π2absq .units
57. The area bounded by the ellipse 4 x2+9 y2=36 is
a) π sq .units b) 3𝜋 sq. unitsc) 6 π sq .units d) 18π sq .units
58. The area bounded by the ellipse 9x2+16 y2=144 is
a) 3 π sq .units b) 4 π sq . unitsc) 6 π sq .units d) 12π sq .units
59. The area bounded by the ellipse x2
25+ y2
16=1 is
a) 5 π sq .units b) 10π sq .unitsc) 15π sq .units d) 20 π sq .units
60. The area bounded by the ellipse x2
a2+ y
2
b2=1 and chord AB where A≡(a ,0) and B
≡(0 , b) isa)ab (π−2 )2
sq . units b)ab(π−2)
4sq .units
c)ab (π−4 )
2sq .units d)
ab(π−4)4
sq . units
61. The area bounded by the ellipse b2 x2+a2 y2=a2b2 and the ordinates x=0,x=ae,(e<1)is
a) ab (e √e2−1+sin−1e )b) ab (e√e2−1−sin−1e )c) ab (e√1−e2+sin−1 e )d) ab (e √1−e2−sin−1 e)
62. The area of region bounded by the parabola y2=4ax and the line y=x isa)8a2
3sq .units b)
4 a2
3sq .units
c)2a2
3sq .units d)
a2
3sq .units
63. The area of region bounded by the parabola y2=4 x and the line y=x isa)163sq .units b)
83sq .units
c)43sq .units d)
23sq .units
64. The area of region bounded by the parabola y2=4 x and the line y=2x isa)83sq .units b)
43sq .units
c)23sq .units d)
13sq .units
65. The area of region bounded by the parabola y=x2 and the line y=x is [U.P.S.E.A.T. 2004]
a)12sq .units b)
13sq .units
c)14sq .units d)
16sq .units
66. The area of region bounded by the parabola y2=4 x and the line x+ y=2 isP a g e |4
a)12sq .units b)
32sq .units
c)72sq .units d)
92sq .units
67. The area of region bounded by the parabola y2=2 x and the line y=4 x−1isa)316
sq .units b)332sq .units
c)916
sq .units d)932sq .units
68. The area of region bounded by the parabola y2=4 x and the line 3 y=2 x+4isa)12sq .units b)
13sq .units
c)23sq .units d)
32sq .units
69. The area of region bounded by the parabola y2=x,X-axis and the line x+ y=2 in the first quadrant isa)72sq .units b)
73sq .units
c)75sq .units d)
76sq .units
70. The area of region bounded by the parabola y2=16 x,and the chord AB,where A≡(1,4) and B≡(9,12)
a)23sq .units b)
43sq .units
c)83sq .units d)
163sq .units
71. The area of region bounded by the parabola 4 y=3 x2 and the line 2y=3 x+12isa) 27 sq.units b) 18 sq.unitsc) 9 sq.units d) 3 sq.units
72. The area of region bounded by the parabola x2= y and the line y=x+2 and X-axisisa)32sq .units b)
23sq .units
c)92sq .units d)
29sq .units
73. The area of region bounded by the parabola y=x2+1 and the line y=2¿) and X-axisisa)23sq .units b)
43sq .units
c)83sq .units d)
163sq .units
74. The area of region bounded by the parabola y=x2+1 and the line y=x , x=0 , x=2 isa)83sq .units b)
43sq .units
c)23sq .units d)
13sq .units
75. The area of region bounded by the parabola y=x2+2 and the line y=x , x=0 , x=3 isa)72sq .units b)
172sq .units
c)212sq .units d)
152sq .units
76. The area of region bounded by the parabola y=x2+1 and the line y=x+1 isa)12sq .units b)
13sq .units
c)16sq . units d)
112sq .units
77. The area of region bounded by the parabola y=x2−6x+11 and the line y=5−x isa)12sq .units b)
13sq .units
c)16sq . units d)
112sq .units
78. The area of region bounded by the parabola y2=4ax and x2=4 ay isa) 2a
2
3sq .units b) 4 a
2
3sq .units
c) 8a2
3sq .units d) 16a
2
3sq .units
79. The area of region bounded by the parabola y2=x and x2= y is [M.P.C.E.T.1997]
a)163sq .units b)
83sq .units
c)13sq .units d)
43sq .units
80. The area of region bounded by the parabola y2=4 x and x2=4 y is [Karnataka C.E.T 1999,2003]
a)23sq .units b)
43sq .units
c)83sq .units d)
163sq .units
81. The area of region bounded by the parabola y2=12 x and x2=12 y isa) 3 sq.units b) 4 sq.unitsc) 36 sq.units d) 48 sq.units
P a g e |5
82. The area of region bounded by the parabola 4 y2=9 x and 3 x2=16 y isa) 4 sq.units b) 2 sq.unitsc) 6 sq.units d) 3 sq.units
83. The area of region bounded by the parabola y2=x+1 and y2=−x+1 isa)23sq .units b)
43sq .units
c)83sq .units d)
163sq .units
84. The area of the region in the first quadrant enclosed by the circle x2+ y2=32 and the line y=x isa) π sq .units b) 2π sq .unitsc) 3 π sq .units d) 4 π sq . units
85. The area of the region enclosed by the circle x2+ y2=36 and the line x+ y=6 isa) 9 π+18 sq .units b) 9 π−18 sq .unitsc) 3 π+6 sq .units d) 3 π−6 sq .units
86. The area of the smaller region bounded by the ellipse 4 x2+9 y2=36 and the line 2 x+3 y=6 is
a)32
(π−1 ) sq .units b)32
(π−2 ) sq .units
c)34
(π−1 ) sq .units d)34
(π−2 ) sq .units
87. The area bounded by the curves y=x2+5 and y=x3∧thelines x=1 , x=2 isa)3112sq .units b)
4312
sq .units
c)316sq .units d)
436sq .units
88. The area of the triangular region bounded by the line2 y=x , y=2x and x=4 isa) 3 sq.units b) 6 sq.unitsc) 12 sq.units d) 24 sq.units
89. The area of the triangular region bounded by the liney=2x+1 , y=3x+1 and x=4 isa) 2 sq.units b) 4 sq.unitsc) 8 sq.units d) 16 sq.units
90. The area of ∆ ABC with vertices A(2,1),B(3,1) and C(5,7) isa) 5 sq.units b) 10 sq.unitsc) 15 sq.units d) 20 sq.units
91. The area of ∆ ABC with vertices A(2,1),B(3,4) and C(5,2) isa) 1 sq.units b) 2 sq.unitsc) 3 sq.units d) 4 sq.units
92. The area of ∆ ABC with vertices A(2,0),B(4,5) and C(6,3) isa) 3 sq.units b) 4 sq.unitsc) 7sq.units d) 8 sq.units
93. The area of the region enclosed by the curves y=sin x andy=cos x from x=0¿ x= π2 and X-axis is
a) 2−√2 sq .units b) 1−√2 sq .unitsc) 2−2√2 sq .units d) 1−2√2 sq .units
94. The area of the region enclosed by the curves y=sin x andy=cos x and Y-axis isa) 1−√2 sq .units b) 2−√2 sq .unitsc) −1+√2 sq .units d) −2+√2 sq .units
95. The volume of solid generated by revolving the area bounded by the line y=x , y=0 , y=2 and Y-axis about Y-axis is
a)2π3cu .units b)
4 π3cu .units
c)8π3cu .units d)
16π3
cu .units
96. The volume of solid generated by revolving the area bounded by the line y=x , y=0 , x=4 about X-axis is
a)64 π3
cu .units b)32π3
cu .units
c)16π3
cu .units d)8π3cu .units
97. The volume of solid generated by revolving the area bounded by the line y=2x , x=0 , x=3 and X-axis about X-axis is
a) 9 π cu .units b) 18π cu .unitsc) 27 π cu .units d) 36 π cu .units
98. The volume of solid generated by revolving the area bounded by the line y=x+1 , X−axis∧the lines x=0 , x=2 about x-axis is
a)13π3
cu .units b)26π3
cu .units
c)10π3
cu .units d)20π3
cu .units
99. The volume of solid generated by revolving the area bounded by the curves xy=2 , Y−axis∧lines y=1, y=4 about Y-axis is
a) π cu .units b) 2π cu .unitsc) 3 π cu .units d) 4 π cu .units
P a g e |6
100. The volume of solid generated by revolving the area bounded by the curves xy=1 ,∧lines x=1 , x=2 about X-axis is
a)π2cu .units b)
3π2cu .units
c) π cu .units d) 3 π cu .units101. The volume of solid generated by revolving the area bounded by the curves y=simx ,¿x=0 , x= π
2 about X-axis isa)π2cu .units b)
π4cu .units
c) π2
2cu .units d) π
2
4cu .units
102. The volume of solid generated by revolving the area bounded by the parabola y2=4ax,X-axis and its latus rectum about X-axis is
a) π a3cu .units b) 2π a3cu .unitsc) 3 π a3 cu .units d) 4 π a3cu .units
103. The volume of solid generated by revolving the area bounded by the parabola y2=16 x,X-axis and its latus rectum about X-axis is
a) 128π cu .units b) 64 π cu .unitsc) 32π cu .units d) 16 π cu .units
104. The volume of solid generated by revolving the area bounded by the parabola y2=16 x,and the lines x=0 , x=2 about X-axis is
a) 8 π cu .units b) 16 π cu .unitsc) 24 π cu .units d) 32π cu . units
105. The volume of solid generated by revolving the area bounded by the parabola y2=4 x,and the lines x= y about X-axis is
a)32π3
cu .units b)16π3
cu .units
c)8π3cu .units d)
4 π3cu .units
106. The volume of solid generated by revolving the area bounded by the parabola y2=4 x,and the lines y¿2 x about X-axis is
a)π2cu .units b)
π3cu .units
c)2π3cu .units d)
3π2cu .units
107. The volume of solid generated by revolving the area bounded by the parabola y2=3 x,and the lines y¿ x about X-axis is
a)3π2cu .units b)
9π2cu .units
c)27π2
cu .units d)81π2
cu .units
108. The volume of solid generated by revolving the area bounded by the parabola x2= y,and the lines y¿ x about X-axis is
a)π5cu .units b)
2π5cu .units
c)π15cu . units d)
2π15
cu .units
109. The volume of solid generated by revolving the area bounded by the parabola x2= y,and the lines y¿2 x about X-axis is
a)4 π15
cu .units b)16π15
cu .units
c)64 π15
cu .units d)256π15
cu .units
110. The volume of solid generated by revolving the area bounded by the parabola x2=4 y,X-axis and the lines x¿−3and x=4 about X-axis is
a)1267π80
cu .units b)781π80
cu .units
c)1267π20
cu .units d)781π20
cu .units
111. The volume of solid generated by revolving the area bounded by the parabola y2=16 x,and chord AB,A≡ (1,4 ) ,B≡(9,12) about X-axis is
a)4 π3cu .units b)
16π3
cu .units
c)64 π3
cu .units d)256π3
cu .units
112. The volume of solid generated by revolving the area bounded by the parabola y2=4 x,and x2=4 y,about X-axis is
a)96π5
cu .units b)48 π5
cu .units
c)24 π5
cu .units d)12π5
cu .units
113. The volume of solid generated by revolving the area bounded by the parabola y2=x,and x2= y,about X-axis is
a)π5cu .units b)
3π5cu .units
c)π15cu . units d)
3π15
cu .units
P a g e |7
114. The volume of solid generated by revolving the area bounded by the circle x2+ y2=25,and the line 3 x=4 y and X-axis in first quadrant is
a)5π3cu .units b)
10π3
cu .units
c)25π3
cu .units d)50π3
cu .units
115. The volume of solid generated by revolving the area bounded by the circle x2+ y2=36,and the line x+ y=6 and X-axis in first quadrant is
a) 18π cu .units b) 36 π cu .unitsc) 54 π cu .units d) 72π cu .units
116. The volume of solid generated by revolving the area bounded by the circle x2+ y2=25,and the line x=0 , x=3 and X-axis is
a) 26 π cu .units b) 38 π cu .unitsc) 52π cu .units d) 66 π cu .units
117. The volume of solid generated by revolving the area bounded by the circle x2+ y2=4,and the parabolay2=3 x and X-axis in first quadrant about X-axis is
a)19π2
cu .units b)19π3
cu .units
c)19π6
cu .units d)19π12
cu .units
118. The volume of solid generated by revolving the area bounded by the ellipse x2
a2+ y
2
b2=1 about major axis is
a) 2π a2b3
cu .units b) 4 π a2b3
cu .units
c) 2π b2a3
cu .units d) 4 π b2a3
cu .units
119. The volume of solid generated by revolving the area bounded by the ellipse 9 x2+16 y2=144 about major axis is
a) 12π cu .units b) 24 π cu .unitsc) 36 π cu .units d) 48 π cu .units
120. The volume of solid generated by revolving the area bounded by the ellipse x2
16+ y2
9=1 about major axis is
a) 32π cu .units b) 64 π cu .units
c)72π3
cu .units d)144π3
cu .units
121. The volume of solid generated by revolving the area bounded by the ellipse 9 x2+4 y2=36 about Y-axis is
a) 2π cu .units b) 4 π cu .unitsc) 8 π cu .units d) 16 π cu .units
122. The volume of solid generated by revolving the area bounded by the arc AB and chord AB of ellipse x2
a2+ y
2
b2=1 with AA’ as
major axis and BB’ as minor axis isa) πab
2
3cu .units b) π a
2b3
cu . units
c) πab2
6cu .units d) π a
2b6
cu . units
123. The volume of solid generated by revolving the area bounded by the one branch of rectangular hyperbola x2− y2=a2 about X-axis isa)π a3
3cu .units b)
π a3
2cu .units
c)4 π a3
3cu .units d)
2π a3
3cu .units
124. The volume of sphere of radius r isa)π r3
3cu .units b)
π r3
2cu .units
c)4 π r3
3cu .units d)
2π r3
3cu .units
125. The volume of sphere of radius 4 isa)256π3
cu .units b)128π3
cu .units
c)64 π3
cu .units d)32π3
cu .units
126. The volume of right circular cylinder of base radius r and height h isa) πrhcu .units b) 2πrhcu .unitsc) π r2hcu .units d) 2π r2hcu .units
127. The volume of right circular one of base radius r and height h isa) π h
2 r2
cu .units b) π h2 r3
cu .units
c) π r2h2
cu .units d)
128. The area bounde by the X-axis and The curve y=sin x and x=0 , x=π is [Kerala P.E.T. 2002]
a) 1 sq.units b) 2 sq.unitsc) 3 sq.units d) 4 sq.units
P a g e |8
129. The area bounded by the curve y2=x,line y=4,Y-axis is [Roorkee 1995,Rajastan P.E.T.2003]
a)163sq .units b)
643sq .units
c) 7√2 sq .units d) 8√2 sq .units130. Area bounded by the curve y=x2,X-axis and linex=1,is
a)13sq .units b)
12sq .units
c) 1 sq .units d) None of these131. Area bounded by the curve y=|x|, and y¿4−|x|,is
a) 4 sq.units b) 16 sq.unitsc) 2 sq.units d) 8 sq.units
132. Area bounded by x=1 , x=2 , xy=1 and X-axis isa) ( log 2 ) sq .units b) 2 sq .unitsc) 1 sq.units d) None of these
133. Area bounded by the curve y=2x−x2, and X-axis isa)23sq .units b) 1 sq.units
c) 2 sq.units d)43sq .units
134. Area bounded by the curve y2=16 x, and line y=mx is 23 ,then m is equal to
a) 3 b) 4c) 1 d) 2
135. Area bounde by the curves y2=x∧x2= y isa)23sq .units b) 1 sq.units
c)12sq .units d) None of these
136. Area of the smaller portion bounded by the ellipse 16 x2+25 y2=400 and line 4 x+5 y=20 isa) 20 π sq .units b) 5 (π−2 ) sq .unitsc) 10 (π−2 ) sq .units d) None of these
137. Area enclosed within the curve |x|+|y|=1 isa) 1 sq.units b) 2 sq.unitsc) 4 sq.units d) None of these
138. The area bounded by Y-axis,y=cos x∧ y=sin x , x ≥0 is
a) 2 (√2−1 ) sq .units b) (√2−1 ) sq .unitsc) (√2+1 ) sq .units d) √2 sq .units
139. Area enclosed by y=1∧±2 x+ y=2 (in sq.unts) isa)12 b)
14
c) 1 d) None of these140. Area common to curve y=x3∧ y=√ x is
a)512 b)
53
c)54 d) None of these
141. The area of the region bounded by y=|x−1|∧¿y=1 is
a) 1 b) 2c)12 d) None of these
142. The area of the region bounded by the curve y2=4 x Y-axis and the line y=3 is
a) 2 sq.units b)94sq .units
c) 6√3 sq .units d) None of these143. The area bounded by the curve y ,=|x|,y=|x−1|∧¿ X-axis is
a) 1 sq.units b)12sq .units
c)14sq .units d) None of these
144. Area of rhombus enclosed by the lines ax ±by±c=0 is
a)12sq .units b)
12c2
absq .units
c) 2c2
absq .units d) None of these
145. The area of the quadrilateral formed by the tangents as the end points of the latus rectum to the ellipse x2
9+ y
2
5=1 is
a)274sq .units b) 9 sq.units
c)272sq .units d) None of these
146. The area bounded by y=log x,X-axis and ordinates x=1 , x=2 isa)12
(log 2 )2 sq . units b) log2esq .units
P a g e |9
c) log 4esq .units d) None of these
147. The area bounded by y=x,X-axis and ordinates x=−1 , x=2 isa) 0 sq.units b)
12sq .units
c)32sq .unit d)
52sq .units
148. The area between the parabola y=x2 and the line y=x isa)16sq .units b)
13sq .units
c)12sq .units d) None of these
149. The area of region bounded by the curvesy=|x−2|, x=1, x=3 and X-axis isa) 1 sq.units b) 2 sq.unitsc) 3 sq.units d) 4 sq.units
150. The area bounded by the curves y=√x ,2 y+3=x and X-axis in Ist Quadrant is
a) 9 sq.units b)274sq .units
c) 36 sq.units d) 18 sq.units151. Area bounded by the curve y=sin x and the linesx=0 , x=π
2∧x=−π
2 isa) 2 b) 4c) 8 d) 16
152. Volume of solid obtained by the complete revolution of the ellipse x2
a2+ y
2
b2=1 , a>b ,
About major axis isa)43π a3 b)
43π b3
c)43π ab2 d)
43π a2b
153. Volume of solid surface generated by rotating the curve y=4 x2 from(0,0 )¿(1,4 ) about Y-axis isa) π cu .units b) 2π cu .unitsc) 4 π cu . units d) 8 π cu .units
154. The volume of the solid generated by revolving the area enclosed by y=4 x2 , x=0∧ y=16About Y-axis is
a) 16 π cu .units b) 32π cu . unitsc) 64 π cu .units d) 48 π cu .units
155. The volume of the solid generated by revolving the area boundedby the parabola x2=4 y,X-axisThe line x=−3∧x=4 about X-axis is
a) 15.783π c .u . b) 15.873π c .u .c) 15.837 π c .u . d) 15.738π c .u .
156. The volume of the solid generated by revolving the region bounded by the curve x= 2
y ,Y-axis x=0The line y=1∧ y=4 about Y-axis isa) 2π cu .units b) 3 π cu .unitsc) 4 π cu .units d) 5 π cu .units
157. The triangle bounded by the lines y=0 , y=x∧¿x=0 is revolved X-axis .Then the volume of the solid thus generated is
a)64 π3 b)
16π3
c)12π3 d)
4 π3
158. Area enclosed between the curves y=x1 /3 ,the Y-axis and the lines y=−1, y=1 isa) 0 b)
12
c)32 d)
33
159. The area of the triangle formed by the lines y=2x , x=0∧ y=2 isa) 1 sq.units b) 2 sq.unitsc) 3 sq.units d) 4 sq.units
160. The area of the region bounded by the curve y2=16 x and lines x=4 , x=1 above the X-axis isa)365sq .units b)
473sq .units
c)563sq .units d)
597sq .units
161. The area between parabola y2=4 x and its latus rectum isa)23sq .units b)
83sq .units
c)163sq .units d)
323sq .units
162. The area bounded by y=sin x,the X-axis and the ordinates x=0 , x=2π ,isa) 4 sq.units b) 6 sq.unitsc) 8 sq.units d) 10 sq.units
P a g e |10
163. The area of region bounded by the curves y=x2+2 , y=x , x=0∧x=3is
a) 8.5 sq.units b) 9.5 sq.unitsc) 10.5 sq.units d) 11.5 sq.units
164. The area enclosed by two curves y2=x+1∧¿y2=−x+1,is
a)23sq .units b)
35sq .units
c)43sq .units d)
83sq .units
165. The area of the region bounded by the parabola y=4 x2 the Y-axis ,y=1∧ y=4,lying in the first quadrant isa)83sq .units b)
73sq .units
c)53sq .units d)
13sq .units
166. The area bounded by the curve y=log x,the axis of X and the line x=a , x=b isa) a log( be )+b log( ae ) b) a log( be )−b log( ae )c) −a log( be )+b log( ae ) d) None of theses
167. The area bounded by the curve y=cos x¿0¿ π,is
a) 2 sq.units b) 3 sq.unitsc) 4 sq.units d) 5 sq.units
168. The area of region in the first quadrant bounded by the circle x2+ y2=32,the line y=x and X-axis isa) 4 π sq .units b) 3 π sq .unitsc) 2π sq .units d) 1π sq .units
169. The volume of the solid generated by the revolving the region bounded by the parabola y2=16 x and its latus rectum is about X-axis is
a) 120 sq.units b) 125 sq.unitsc) 128 sq.units d) 132 sq.units
170. The volume obtained by the revolving the region bounded by x2− y2=9 , y=0 and its latus rectum is about X-axis isa) 18π (2−√2 )c .u b) 20 π (2−√2 )c .uc) 22π (2−√2 )c .u d) 32π (2−√2 ) c .u
171. Find the volume of solid generated by the revolving the region bounded by the line y=2x,X-axis and the lines x=0 , x=3
a) 26 π cu .units b) 36 π cu .unitsc) 39π cu .units d) 42 π cu .units
172. Volume of solid generated by the revolving the curve about X-axis ,if the area enclosed by the parabola y2=4ax and its latus rectum isa) 4 πacub .units b) 4 π a2cub .unitsc) 4 π a3cub .units d) None of these
173. Find the area of the triangular region whose side have the equationy=2x+1 , y=3x+1∧¿x=4
a) 6 sq.units b) 8 sq.unitsc) 10 sq.units d) 12 sq.units
174. The circle x2+ y2=a is revolved about X-axis.Find the volume of sphere so formeda)43π a2 cu .units b)
43πacu .units
c)43π a3 cu .units d) None of these
175. The area bounded by the curve y=x,X-axis and lines x=−1 , x=2,is given bya) 2/3 b) 5/2c) 7/4 d) 9/11
176. The area of ellipse x2a2
+ y2
b2=1,is given by
a) πab b) 2πab
c)π2ab d)
23πab
177. The line y=x+1 is revolved about X-axis.The volume of solid of revolution formed by revolving the area of covered by the given curve,x-axis and the lines x=0 , x=2,isa)13π3 b)
17π3
c)19π3 d)
26π3
178. The surface of the sphere obtained by revolving the circle x=r . cosθabout X-axis is equal yoa) π r2 b) 2π r2
c) 4 π r2 d)43πr 2
P a g e |11
179. Find the volume of the solid formed when the area between the X-axis ,the line x=2 , x=4 andThe curve y=x2 is rotated once about the X-axis.Leaves your answer as multiple of π.
a)7723
π cu .units b)8825
π cu .units
c)9923
π cu .units d) None of these180. If the portion of the parabola y2=4 x lying between the vertex and the latus rectum is revolved about X-axis, then the volume of the solid thus generated is
a)π2 b) π
c) 2π d) π2
181. The area bounded by the curve y=log x,X-axis and the ordinates x=1∧x=e is a) e sq .units b) 2e sq .unitsc) log e sq .units d) 2 log e sq .units
182. The area bounded by the curve y2=4 x and the line x=1 , x=4 above X-axis isa)73sq .units b)
143sq .units
c)283sq .units d)
563sq .units
183. The area of the region bounded by the parabolay2=x2+1 and the line y=x+1 is
a)12sq .units b)
13sq .units
c)16sq .units d)
112sq .units
184. The area bounded by the parabolay2=4ax andIts latus rectum is
a)4 a2
3sq .units b)
8a2
3sq .units
c)4 a√a3
sq .units d)8a√a3
sq .units
185. The area of triangular region bounded by the linesy=2x+1 , y=3x+1∧x=4is
a) 2 sq.units b) 4 sq.unitsc) 8 sq.units d) 16 sq.units
186. The area bounded by the parabola y=x2+3,
X-axis and the lines x=0 , x=3 isa) 3 sq.units b) 6 sq.unitsc) 12 sq.units d) 18 sq.units
187. The area bounded by the line y=x,X-axis and theOrdinates x=−1 , x=2 isa)52sq .units b)
32sq .units
c)13sq .units d)
23sq .units
188. The area bounded by the ellipse x2
a2+ y
2
b2=1 and
Chord AB where A≡(a ,o) and B≡(0 , b) isa)ab (π−2 )2
sq . units b)ab(π−2)
4sq .units
c)ab (π−4 )
2sq .units d)
ab(π−4)4
sq . units
189. The area of the region bounded by the parabolay2=4ax and the line y=mx is
a)8 a2
3m3sq .units b)
4 a2
3m3sq .unitsl
c)2a2
3m3sq .units d)
a2
3m3sq .units
190. The area bounded by the curve x=at 2 , y=2atBetween the ordinates corresponding to t=1∧t=2 is
a) 7a2
3sq .units b) 14a
2
3sq .units
c)28a2
3sq .units d)
56a2
3sq .units
191. The area bounded by the curve y=sin2 x ,X-axisAnd the lines x=π
4∧x=3 π
4 isa) 1 sq.units b) 2 sq.unitsc)12sq .units d)
14sq .units
192. The area bounded by the curve y2=kx,X-axis and the ordinates x=c isa)c3 √kc sq .units b)
2c3 √kc sq .units
c)4 c3 √kc sq .units d)
8c3 √kc sq .units
193. The area bounded by the line y=x3,X-axis and the ordinates x=1 , x=3 isP a g e |12
a) 5 sq.units b) 10 sq.unitsc) 20 sq.units d) 40 sq.units
194. The area bounded by the curve y=cos3 x ,
0≤ x≤ π6 is
a)13sq .units b)
23sq .units
c)16sq .units d)
32sq .units
195. The area enclosed by the circle of radius r is a) πr sq .units b) 2πr sq .unitslc) π r2 sq .unitsl d) 2π r2 sq .units
196. The volume of solid generated by revolving the area bounded by the lines y=x , y=0 , y=2 and Y-axis is about Y-axis is
a)2π3cu .units b)
4 π3cu .units
c)8π3cu .units d)
16π3
cu .units
197. The volume of right circular cylinder of base radius r and height h isa) πrhcu .units b) 2πrhcu .unitsc) π r2hcu .units d) 2π r2hcu .units
198. The volume of right circular one of base radius r and height h isa)πr h2
2cu .units b)
πr h2
3cu .units
c)π r2h2
cu .units d)π r2h3
cu .unitsl
199. The volume of solid generated by revolving the area bounded by the lines y=2 x , x=0 , x=3 and X-axis about X-axis is
a) 9 π cu .units b) 18π cu .unitsc) 27 π cu .units d) 36 π cu .units
200. The volume of solid generated by revolving the area bounded by the parabola x2=4 y,X-axis and lines x=−3∧x=4 about X-axis is
a)1267π80
cu .units b)781π80
cu .units
c)1267π20
cu .units d)781π20
cu .units
201. The volume of solid generated by revolving the area bounded by the circle x2+ y2=25 ,th lines x=0 , x=3 about X-axis is
a) 26 π cu .units b) 38 π cu .unitsc) 52π cu .units d) 66 π cu .units
202. The volume of solid generated by revolving the area bounded by the one branch of rectangular hyperbola x2− y2=a2 line x=2a about X-axis isa)π a3
3cu .units b)
π a3
2cu .units
c)4 π a3
3cu .units d)
2π a3
3cu .units
203. The volume of solid generated by revolving the area bounded by the curve y=sin x ¿ x=0¿ x=π
2 cu.units about X-axis isa)π2sq .units b)
π4sq .units
c) π2
2sq .units d) π
2
4sq .units
204. The volume of solid generated by revolving the area bounded by the circle x2+ y2=4 and parabola 2 y=3 x and X-axis in first quadrant about X-axis is
a)19π2
cu .units b)19π3
cu .units
c)19π6
cu .units d)19π12
cu .units
Application of Definite Integrals205. The volume of solid generated by revolving the area bounded by the curve xy=1 ,and the linesx=1∧x=4 about X-axi is
a)π2cu .units b)
3π2cu .units
c) π cu .units d)3π4cu .units
P a g e |13
SAMEJ TUTORIALSMT-CET 2013
Date : TEST ID: 33Time : 06:09:00 Hrs. MATHEMATICS-IIMarks : 410
7) APPLICATIONS OF DEFINITE INTEGRAL
: ANSWER KEY :
1) c 2) c 3) d 4) a5) a 6) d 7) c 8) c9) c 10) d 11) a 12) a13) d 14) d 15) d 16) d17) a 18) b 19) a 20) a21) b 22) b 23) c 24) a25) d 26) b 27) b 28) d29) a 30) c 31) b 32) d33) d 34) a 35) a 36) b37) d 38) a 39) c 40) c41) d 42) d 43) a 44) d45) d 46) c 47) d 48) d49) d 50) c 51) c 52) c53) d 54) d 55) b 56) a57) c 58) d 59) d 60) b61) c 62) a 63) b 64) d65) d 66) d 67) d 68) b69) d 70) d 71) a 72) c73) b 74) a 75) c 76) c77) c 78) d 79) c 80) d81) d 82) a 83) c 84) d85) b 86) a 87) b 88) c89) c 90) b 91) d 92) c93) a 94) c 95) c 96) a97) d 98) b 99) c 100) a101) d 102) b 103) a 104) d
105) a 106) c 107) b 108) d109) c 110) a 111) d 112) a113) d 114) d 115) d 116) d117) c 118) d 119) d 120) d121) d 122) a 123) c 124) c125) a 126) c 127) d 128) d129) b 130) a 131) d 132) a133) d 134) b 135) d 136) b137) b 138) b 139) a 140) a141) a 142) b 143) c 144) c145) d 146) c 147) d 148) a149) a 150) a 151) a 152) c153) b 154) b 155) c 156) b157) a 158) b 159) a 160) c161) b 162) a 163) c 164) d165) b 166) c 167) a 168) a169) c 170) a 171) b 172) c173) b 174) c 175) b 176) a177) d 178) c 179) c 180) c181) c 182) c 183) c 184) b185) c 186) d 187) a 188) b189) a 190) d 191) a 192) b193) c 194) a 195) c 196) c197) c 198) d 199) d 200) a201) d 202) c 203) d 204) c205) d
P a g e |14
SAMEJ TUTORIALSMT-CET 2013
Date : TEST ID: 33Time : 06:09:00 Hrs. MATHEMATICS-IIMarks : 410
7) APPLICATIONS OF DEFINITE INTEGRAL
: HINTS AND SOLUTIONS :
1 (c)The bounded area is shown in figureRequired area isA=∫
a
b
ydx
A=∫a
b
mxdx
A=[mx22 ]baA=
m (b2−a2 )2
sq .units
2 (c)
Area=∫1
2
2xdx
Area=[ x2 ]21
Area=3Sq .units
3 (d)
Area=∫1
3
3xdx
Area=32
[ x2 ]31
Area=12Sq .Units
4 (a)The bounded area is shown in figureRequired area isA=|∫
−1
0
ydx|+∫02
ydx
A=|∫−1
0
xdx|+∫02
xdx
A=|[ x22 ] 0−1|+[ x22 ]20A=1
2+2
A=52Sq .units
5 (a)The bounded area is shown in figureRequired area isA=∫
a
b
ydx
P a g e |15
A=∫a
b
(mx+c)dx
A=[mx22 +cx ]baA=
m (b2−a2 )2
+c (b−a)sq .units
6 (d)
Area=∫2
5
(2 x+3)dx
Area=[ x2+3 x ]52
Area=30Sq .units
7 (c)
Area=12∫28
(5 x+7)dx
Area=12 {52 x2+7 x}82
Area=12 {(160+56−24 ) }
Area=96Sq .units
8 (c)The bounded area is shown in figure
Required area isA=|∫
−4
1
( x2−1)dx|A=|[ x24 −x ]−1−4|A=|14 +1−4−4|A=27
4Squnits
9 (c)The bounded area is shown in figureRequired area isA=∫
1
3
x3dx
A=[ x 44 ]31A=1
4[81−1]
A=20Sq .units
10 (d)Required area isA=∫
a
b
ydx
A=∫1
3
( 4x2 )dxA=[−4x ]31A=−4
3+4
P a g e |16
A=83Sq .unts
11 (a)Required area isA=∫
0
1
2√1−x2dx
A=2[ x2 √1−x2+12sin−1 x]10
A=0+sin−1 (1 )−0−sin−10
A=π2sq .units
12 (a)Required area isA=∫
7
14 dx√ x2+6 x−55
A=∫7
14 dx
√ ( x+3 )2−82
A=[ log|( x+3 )+√( x+3 )2−82|]147
A=log|17+√196+84−55|−log|10+√49+42−55|A=log 32−log16A=log 2 sq. units
13 (d)The bounded area is shown in fig.The curve isy2=x2(1−x )Put y=0x2 (1−x )=0x=0 , x=1A=(1,0)Required Area isA=2∫
0
1
x √1−xdx
By∫0
a
f ( x )dx=¿∫0
a
f (a−x)dx¿
A=2∫0
1
(1−x )√ xdx
A=2[2 x√ x3 −2 xx √x5 ]10
A=43−45
A= 815
sq .units
P a g e |17
14 (d)
Area=∫1
4 2xdx
Area=2 [ log x ] 41
Area=4 log2 sq .units
15 (d)
Area=∫a
b
ydx
Area=∫1
3
( 4x )dxArea=4 [ log x ]3
1Area=4 log3 sq .units
16 (d)
Area=∫4
8
( 16x )dxArea=16 [ log x ] 8
4Area=16 log 2 sq .units
17 (a)
Area=∫2
8
( 12x )dxArea=1
2[ log x ] 8
2Area=log 2 sq .units
18 (b)The bounded area is shown in fig.The curve isA=2 AreaOAB
A=2∫0
π
sin x dx
A=2 [−cos x ]π0
A=2(−cos π+cos 0)A=4 sq . units
19 (a)The bounded area is shown in fig.The curve is
P a g e |18
A=∫π4
π2
ydx+∫π2
3 π4
− ydx
A=∫π4
π2
sin 2xdx−∫π2
3 π4
sin 2 xdx
A=[−cos2 x2 ]π2
π4−[−cos 2x2 ]
3π4π2
A=−12
[cos π−cos π2−cos 3π
2+cosπ ]
A=−12
[−1−0−0−1]
A=1Sq .units
20 (a)
Area=∫0
π2
cos xdx
Area=[sin x ]π20
Area=1Sq .unts
21 (b)
Area=∫0
π2
(cos x )dx+∫π2
π
|cos x|dx
Area=[sin x ]
π2
0+|[sin x ] ππ2 |Area=1+|1|Area=2 sq .units
22 (b)
Area=4∫0
π2
(cos x )dx
Area=4 [sin x ]π20
Area=4 sq .units
23 (c)The bounded area is shown in fig.The curve isA=4 AreaOAB
A=4∫0
π2
ydx
A=4∫0
π2
2cos xdx
A=8[sin x ]π20
A=8[sin( π2 )−sin 0]P a g e |19
A=8 sq .units 24 (a)The bounded area is shown in fig.The curve isA=∫
0
π6
cos3 xdx
A=[ sin3 x3 ] π60
A=13 [sin( π2 )−sin 0]
A=13sq .units
P a g e |20
25 (d)Required Area isA=∫
0
π2
sin2( x2 )dx
A=∫0
π2
( 1−cos x2 )dxA=1
2[x−sin x ]
π20
A=12 [ π2−sin (π2 )−0]
A=14
(π−2 ) sq .units
26 (b)
Area=∫0
π2
(2 x+sin x )dx
Area=[ x2−cos x ]π20
Area=( π24 −0)−(0−1)
Area=1+ π2
4sq .units
27 (b)The bounded area is shown in fig.The curve isA=2 AreaOSL
A=2∫0
a
2√ax dx
A=4 √a [ 2 x√ x3 ]a0A=8
3 √a[a√a−0]
A=8 a2
3sq .units
28 (d)The bounded area is shown in fig.The curve isA=2 AreaOSL
A=2∫0
2
2√2x dx
A=4 √2[ 4√23 ]
A=323sq .units
29 (a)
Area=2∫0
4
(4 √x )dx
Area=8[ 23 x32 ]40
Area=1283
sq .units
30 (c)The bounded area is shown in fig.The curve isA=∫
1
4
2√x dx
A=[ 4 x √x3 ]41
A= 43[8−1]
P a g e |21
A=283sq .units
31 (b)
Area=∫2
4
2√x dx
Area=2[ x32
32 ] 42
Area=43(8−2√2)
Area=83
(4−√2 ) sq .units
32 (d)
Area=∫4
9
3√x dx
Area=3( 23 ) [ x32 ]94
Area=2(27−8)Area=38 sq .units
33 (d)The bounded area is shown in fig.The curve isA=∫
0
a
ydx
A=∫0
a x2
adx
A=[ x33a ] a0A=a2
3sq .units
34 (a)
Area=∫a
2a x2
adx
Area=[ x33a ]2aaArea=7 a
2
3sq .units
P a g e |22
35 (a)The bounded area is shown in fig.The curve isA=∫
2
4
2√ y dy
A=2[2 y √ y3 ]42
A=43
(4 √4−2√2 )
A=43
(8−2√2 ) sq .units
36 (b)
Area=∫1
4
4 √ y dy
Area=83
[ y 32 ]41
Area=563sq .units
37 (d)
Area=∫1
4 √ y2dy
Area=13
[ y 32 ]41
Area=73sq .units
38 (a)The curve is y=√6 x+4y2=6 x+4
y2=6 (x+ 23 ) represented a parabola with vertex at (−23 ,0),The bounded area is shown in fig.The curve isA=∫
0
2
√6 x+4dx
A=[ 2 (6 x+4 )32
3 (6 ) ]20A=1
9[ (16 )
32−(4 )
32 ]
A=569sq .units
39 (c)The curve is y=4√ x−1y2=16 ( x−1 ) represented a parabola with vertex at (1,0 ),
P a g e |23
The bounded area is shown in fig.The curve isA=∫
1
3
4 √x−1dx
A=4 [ 2 ( x−1 )32
3 ]31A=8
3(2√2−0)
A=16√23
sq .units
40 (c)The curve is y=2x−x2
y=−(x2−2x+1 )+1y−1=−( x−1 )2 represented a parabola with vertex at (1,1 ),The bounded area is shown in fig.The curve isA=∫
0
2
(2 x−x2)dx
A=[ x2− x3
3 ]2oA=4−8
3−0
A=43sq .units
41 (d)The curve is y=4 x−x2
y=−(x2−4 x+4 )+4y−4=−( x−2 )2 represented a parabola with vertex at (2,4 ),The bounded area is shown in fig.The curve isA=∫
0
4
(4 x−x2)dx
A=[2 x2− x3
3 ]4oA=32− 64
3
A=323sq .units
42 (d)The curve isy=16−x2
y−16=−x2 represented a parabola with vertex at (0,16 ),The required area isA=∫
0
4
(16−x2)dx
A=[16 x− x3
3 ]4oA=64−64
3
A=1283
sq .units
43 (a)
P a g e |24
The bounded Area is as shown in figureThe Required area isA=∫
0
2
(4−x2)dx
A=[4 x− x3
3 ]20A=8−8
3−0
A=163sq .units
44 (d)
Area=∫0
3
(x2+3 )dx
Area=[ x33 +3 x]30Area=18 sq .units
45 (d)
Area=∫0
3
(x2+1 )dx
Area=[ x33 +x ]30Area=12 sq .units
46 (c)
Area=2∫0
1
(2−x2 )dx
Area=2[2x− x3
3 ]10Area=2( 53 )Area=10
3sq .units
47 (d)The bounded area is shown as followCurves arey=x2−3 xand y=2xSolving themx=0 , x=5y=0 , y=10A≡ (5,10 ) ,C≡(5,0)Put y=0 inparabola equtionx (x−3 )=0x=0,3B≡(3,0)Required area is A=Area OABA=Area OAC – Area ABCA=∫
0
5
y linedx−∫3
5
y paraboladx
A=∫0
5
2xdx−∫3
5
(x2−3 x)dx
P a g e |25
A=[ x2 ]5
0−[ x33 −3x2
2 ]53
A=25−1253
+ 752
+9−272
A=493sq . units
48 (d)The bounded area is shown in figure of question 47Required area isA=Area OAB +Area ODBArea ODB=493 sq.unitsAreaODB=|∫0
3
(x2−3 x)dx|=|[ x33 −3 x2
2 ]30|AreaODB=|9−272 |=92Area=49
3+ 92
Area=1256
sq .units
49 (d)The curves isx=a t 2 and y=2aty2=4a2t 2
y2=4a (a t2)y2=4ax represent a standard parabolaAt t=1 , x=aAt t=2 , x=4 aThe bounded area is as shown in fig. Required area isA=2∫
a
4a
2√axdx
A=4 √a [ 2 x√ x3 ]4 aaA=8√a
3(4a√4 a−a√a)
A=56 a2
3sq .units
50 (c)
A=∫0
2
ex dx
A=[ex ]20
A=e2−1 sq .units
51 (c)Curve areay= log xx=1 and x=eRequired Area isA=∫
1
e
log xdx
A=¿
A=(e−0 )−[x ]e1
A=e−e+1A=log e sq. units
52 (c)The bounded area is as shown in figEqution of circlex2+ y2=r2Required area isA=4∫
0
r
√r2−x2dx
P a g e |26
A=4 [ x2 √r2−x2+ r2
2sin−1( xa )] ro
A=4 [ r2 √r2−r2+r2
2sin−1 (1 )−0−r 2
2sin−10]r0
A=2r 2( π2 )A=π r2 sq .units
53 (d)
Area=4∫0
3
√9−x2dx
Area=4 { x3 √9−x2+ 92sin−1( x3 )}30
Area=4 {92 ( π2 )}Area=9π sq .units
54 (d)
Area=4∫0
4
√16−x2dx
Area=4 { x4 √16−x2+162sin−1( x4 )}40
Area=4 {8( π2 )}Area=16π sq .units
55 (b)The bounded area is shown in fig.
The curve isx2+ y2=2ax(x−a)2+ y2=a2 represented a circle with center at (a ,0 ),And radius aRequired area isA=∫
0
a
√2ax−x2dxA=∫
0
a
√a2−( x−a )2dx
A= x−a2 [ x2 √a2− (x−a )2+ a
2
2sin−1( x−a
a )]a0A=0+ a
2
2sin−10−a2
2sin−1(−1)
A=π a2
4sq .units
56 (a)The bounded area is shown in fig.Required area isArea = 4 Area OABA=4∫
0
a ba √a2−x2dx
A=4ab [ x2 √a2−x2+ a
2
2sin−1( xa )]a0
A=4ab
[0+ a2
2sin−1 (1 )−0−a2
2sin−1(0)]
A=2ab( π2 )A=πab sq .units
P a g e |27
57 (c)
Area=4∫0
3 23 √9−x2dx
Area=83∫0
3
√32−x2dx
Area=83 {x2 √32−x2+ 9
2sin−1( x3 )}30
Area=83 ( 92 π2 )
Area=6π sq .units
P a g e |28
58 (d)
Area=4∫0
4 34 √32−x2dx
Area=4∫0
4
√42−x2dx
Area=3 {x2 √42−x2+ 162sin−1( x4 )}40
Area=3(8 π2 )Area=12π sq .units
59 (d)
Area=4∫0
5 45 √25−x2dx
Area=165 ∫
0
5
√52−x2dx
Area=165 {x2 √252−x2+25
2sin−1( x5 )}50
Area=165 ( 252 π
2 )Area=20π sq .units
60 (b)The bounded area is as shown in fig.Area of ellipse isπab sq.unitsArea of ellipse in firset quadrant is πab4 sq.unitsArea of OAB=12 ab sq.units
Required area isA=Area of ellipse in first quadrant – Area of OABA=πab
4−ab2
A=ab(π−2)
4sq .units
61 (c)The bounded area is as shown in fig.Required area isArea=2 AreaOCAD
Area=2∫0
ae ba √a2−x2dx
Area=2ba [ x2 √a2−x2+ a
2
2sin−1 x]ae0
Area=2ba ( ae2 √a2−a2e2+ a
2
2sin−1( aea )−0)
Area=ba(a2 e√1−e2+a2 sin−1 e)
Area=ab(e√1−e2+sin−1 e)
62 (a)
y=x∧ y2=4 ax∴ x2=4ax∴ x ( x−4 a )=0∴ x=0 , x=4a
Area=∫0
4a
2√a√ xdx−∫0
4a
xdx
P a g e |29
Area=2√a [ x32
32 ] 4 a
0−[ x22 ]4 a0
Area=323a2−16
2a2
Area=8 a2
3sq .units
63 (b)
Area=∫0
4a
2√ xdx−∫0
4
xdx
Area=43
[ x 32 ] 4
0−[ x22 ]40
Area=323
−162
Area=83sq .units
64 (d)
Area=∫0
1
2√x dx−∫0
1
2xdx
Area=2( 23 )[ x32 ] 10−2[ x
2
2 ]10Area=4
3−1
Area=13sq .units
65 (d)The bounded area is as shown in figCurves arey=x2 and y=xSolvin themA≡(1,1)Required area isA=∫
0
1
(x−x2)dx
A=[ x22 −x3
3 ]11A=1
2−13
A=16sq .units
66 (d)The bounded area is as shown in figCurves arey2=x and 2¿ x+ ySolvin themA≡(1,1) and B≡(4 ,−2)Required area isA=∫
−2
1
(2− y− y2)dy
A=[2 y− y2
2−y3
3 ] 1−2A=2−1
2−13+4+2−8
3
P a g e |30
A=92sq .units 67 (d)
Area=18∫−1
8
1
( y+14 − y2
2 )dy
Area=18∫−1
2
1
(2+2 y−4 y2 )dy
Area=18 {2 y+ y2− 43 y3}
1−12
Area=18 ( 53 + 7
12 )Area= 9
32sq .units
P a g e |31
68 (b)The bounded area is as shown in figCurves arey2=4 x and 3y¿2 x+4Solvin themy2=2 (3 y−4 )y2−6 y+8=0( y−2 ) ( y−4 )=0y=2, y=4x=1,4A≡(1,2) and B≡(4,4)Required area isA=∫
1
4
(√4 x−( 2 x+43 ))dxA=[2( 2x √ x
3 )−13 (x2+4 x )] 41A=1
3[32−(16+16 )−4+5]
A=13sq .units
69 (d)The bounded area is as shown in figCurves arey2=x and 2¿ x+ ySolvin themA≡(1,1) and B≡(1,0) and C≡(2,0)Required area isA=¿Area OAB +Area ABCA=∫
0
1
√x dx+∫1
2
(2−x)dx
A=[ 2x √x3 ]
1
0+[2 x− x2
2 ]21
A=23+4−2−2+ 1
2
A=76sq .units
70 (d)The bounded area is as shown in figCurves arey2=16 xEquation of chord AB isy−412−4
= x−99−1
y−48
= x−98
x− y+3=0Required area isA=∫
1
9
(4 √x−( x+3 ) )dx
A=[ 8x √ x3
−x2
2−3x ]91
A=72−812
−27−83+ 12+3
A=163sq .units
71 (a)The bounded area is as shown in figCurves are4 y=3 x2 and 2y¿3 x+12Solvin them6 x+24=3 x2
P a g e |32
x2−2 x−8=0x=4 ,−2y=12,3Required Area isA=∫
−2
4
( 3 x+122−3 x
2
4 )dxA=[ 12 (3 x22 +12x )− x3
3 ] 4−2A=3
4[16−4 ]+6 [4+2 ]−1
4[64+8]
A=9+36−18A=3 sq .units
72 (c)The bounded area is as shown in figCurves arey=x2 and y¿ x+2Solvin themx2=x+2x2−x−2=0x=−1 , x=2y=1 , y=4A≡(−1,1) and B≡(2,4 )Required area isA=∫
−1
2
(x+2−x2)dx
A=[ x22 +2x− x3
3 ] 2−1A=2+4−8
3−12+2−1
3
A=92sq .units
73 (b)The bounded area is as shown in figCurves arey=x2+1 is parabola vertex at (2,0)y=2 ( x−1 ) is straight lineSolving them4 ( x−1 )2=16 ( x−2 )x2+2 x−1=4 x−8x2−6 x+9=0( x−3 )2=0x=3 , y=4A≡(1,0) and B≡(3,4 ) and C≡(2,0) and D≡(2,0)Required area isA=¿Area ABD - Area BCDA=∫
1
3
y linedx−∫2
3
y paraboladx
A=∫1
3
2 ( x−1 )dx−∫2
3
4√ x−2dx
A=[ x2−2x ] 31−4 [ 2 ( x−2 )
32
3 ]32A=9−6−1+2−8
3(1)
A=43sq .units
P a g e |33
74 (a)
Area=∫0
2
(x2+1−x )dx
Area=[ x33 +x− x2
2 ]20Area=8
3+2−2
Area=83sq .units
P a g e |34
75 (c)The bounded area is as shown in figCurves arex2+2= y and x= ySolvin themA≡(3,3) and B≡(3,11) and C≡(0,2)Required area isA=∫
0
3
(x2+2−x )dx
A=[ x33 +2x− x2
2 ]30A=9+6−9
2−0
A=212sq .units
76 (c)
Area=∫0
1
[ x+1−(x2−1)]
Area=∫0
1
(x−x2)dx
Area=[ x22 −x3
3 ]10Area=1
2−13
Area=16sq .units
77 (c)The bounded area is as shown in fig
Curves arey=x2−6 x+11y¿ x2−6 x+9+2
y−2=( x−3 )2 represented parabola with vertex at (3,2)y=5−x is straight lineSolving themx2−6 x+11=5−xx2−5 x+6=0( x−2 ) ( x−3 )=0A≡(2,3) ,B≡(3,2)Required area isA=∫
2
3
((5−x)−(x2−6 x+11) )dx
A=[ 5x22 −x3
3−6 x ]32
A=452
−9−18−10+ 83+12
A=16sq .units
78 (d)
Area=∫0
4a
(2√a√x− x2
4 a )dxArea=2√a ( 23 )[ x
32 ] 4 a
0− 14 a
[ x33 ]4 a0Area=4
3 √a (√4a )3− 112a
(4 a )3
Area=323a2−16
3a2
Area=16 a2
3sq .units
P a g e |35
79 (c)
Area=∫0
1
√x dx−∫0
1
x2dx
Area=23
[ x32 ] 1
0− 13
[ x3 ] 10
Area=23−13
Area=13sq .units
80 (d)
Area=∫0
4
(2√ x− x2
4 )dxArea=2( 23 )[ x
32 ] 4
0− 112
[ x3]40
Area=43
(8 )−6412
Area=163sq .units
81 (d)
Area=∫0
12
¿¿ )dxArea=√12( 23 )[ x
32 ] 12
0− 136
[ x3 ]120
Area=23
(144 )−13(144)
Area=1443
Area=48 sq .units
82 (a)
Parabolaare y2=9 x4
∧x2=16 y3
4 a=94,4b=16
3
a= 916, b=4
3Required area isA=16
3 ( 916 )( 43 )A=4 sq. units
83 (c)The bounded area is as shown in figCurves arey2=x+1 and y2=−x+1Solvin themA≡ (−1,0 ) and B≡ (1,0 ) and C≡ (0,1 ),D≡(0 ,−1)Required area isA=2 ( Area I )+2(Area II )
P a g e |36
A=2∫−1
0
√x+1dx+2∫0
1
√−x+1dx
A=2[2 ( x+1 )32
3 ] 0−1+2[2 ( x−1 )
32
−3 ]10A=4
3[1+0+1]
A=83sq .units
84 (d)Area=¿
Area=∫0
4
xdx+∫4
4√2
√32−x2dx
Area=[ x24 ]4
0+[ x2 √32−x2+ 322sin−1( x
4√2 )]4 √24
Area=8+16sin−1 (1 )−[8+16sin−1( 1√2 )]Area=8+16( π2 )−8−16 ( π4 )Area=4 π sq .units
85 (b)The bounded area is as shown in fig
Curves arex2+ y2=36 and x+ y=6Required area isA=Area of the ¿̊ the first quadrant−Areaof ∆OAB
A=π4
(6 )2−12
(6 )(6)
A=9π−18sq .units
86 (a)The bounded area is shown as followEllipse is x2
9+ y
2
4=1Line is2 x+3 y=6From fig.A≡(3,0),B≡(0,2)Required area is
A=∫0
3
( 23 √9−x2−23
(3−x ))dxA=2
3 [ x2 √9−x2+ 92sin−1( x3 )−3 x+ x
2
2 ]30A=2
3 ( 92 sin−1 (1 )−9+ 92 )
A=32
(π−1 ) sq .units
87 (b)The bounded area is shown as followP a g e |37
The curves arey=x2+5 and y=x3Required area is
A=∫1
2
(x2+5−x3)
A=[ x33 +5 x− x4
4 ]21A=8
3+10−16
4−13−5+ 1
4
A=4312
sq .units
88 (c)The bounded area is shown as followThe curves arey=2x ,2 y=x and 4=xSolving themA≡(4,8) ,B≡(4,2)Required area isA=∫
0
4
(2 x− x2 )dx
A=[ x2− x2
4 ]40A=3
4(16)
A=12 sq .units
89 (c)The bounded area is shown as followThe curves are
y=2x+1 , y=3x+1 and 4=xSolving themA≡(0,1) ,B≡ (4,13 ) , C≡(4,9), D≡(4,0)Required area isA=Area ABCA=¿Area OBAD – Area OACDA=∫
0
4
(3x+1 )dx−∫0
4
(2x+1)dx
A=[ 3x22 +x−x2−x] 40A=[ x22 ]40A=8 sq .units
90 (b)A=A (∆ ABC )
A=∫1
3
( x+92 +2x−7)dx+∫3
5
( x+92 −3 x+8)dxA=1
2∫13
(5 x−5)dx+ 52∫35
(5−x )dx
A=12 [ 5 x22 −5x ] 3
1+ 52[5 x− x2
2 ]53A=1
2 ( 452 −15−52+5)+ 52 (25−252 −15+ 9
2 )A=10 sq .units
91 (d)A=A (∆ ABC )
P a g e |38
A=∫2
3
[ (3 x−5 )−( x+13 )]dx+∫35
[ (−x+7 )−( x+13 )]dxA=1
3∫23
(8 x−16)dx+13∫3
5
(−4 x+20)dx
A=83 [ x22 −2 x ] 3
2−43[ x22 −5 x ]53
A=83 [ 92−4]−43 [ 162 −10]
A=123
A=4 sq . units
92 (c)A=A (∆ ABC )
A=12 |2 0 14 5 16 3 1|
A=12|4−18|
A=12|−14|
A=7 sq .units93 (a)The bounded area is shown as followThe curves are
y=sin x and y=cos xRequired area isA=∫
0
π4
sin xdx+∫π4
π2
cos xdx
A=[−cos x ]π4
0+ [sin x ]
π2π4
A=−1√2
+1+1− 1√2
A=2−√2 sq .units
94 (c)The bounded area is shown as followThe curves arey=sin x and y=cos xSolving them
sin x=cos x
x=π4, y= 1
√2
A≡( π4 , 1√2 )Required area isA=∫
0
π4
(cos x−sin x)dx
A=[sin x+cos x ]π40
A= 1√2
+ 1√2
−0−1
A=√2−1 sq .units
95 (c)The area bounded by the curves as shown in fig.Required volume isV=π∫
0
2
x2dy
V=π∫0
2
y2dy
P a g e |39
V=π [ y33 ]20V=8 π
3cu .units
96 (a)
Volume=π∫0
4
y2dx
Volume=π∫0
4
x2dx
Volume=π3
[x3 ] 40
Volume=64π3
cu .units
97 (d)
Volume=π∫0
3
y2dx
Volume=π∫0
3
4 x2dx
Volume=43π [ x3 ]3
0Volume=36π cu .units
98 (b)
Volume=π∫0
2
( x+1 )2dx
Volume=π3 [ ( x+1 )3 ]2
0
Volume=26 π3
cu .units
99 (c)The area bounded by the curves as shown in fig.Curves are xy=2And y=1 to x=4Required volume isV=π∫
1
4 4y2dy
V=4 π [−1y ]41V=4 π [−14 +1]V=3π cu .units
P a g e |40
100 (a)
Volume=π∫1
2 1x2dx
Volume=π [−1x ]21Volume=π
2cu .units
101 (d)The area bounded by the curves as shown in fig.Curves are y=sin xRequired volume isV=π∫
0
π2
sin2 xdx
V=π∫0
π2
( 1−cos x2 )dxV= π
2 [x−sin 2 x2 ] π20
V= π2 ( π2−sin π
2 )V= π2
4cu .units
P a g e |41
102 (b)The area bounded by the curves as shown in fig.Curves is y2=4axLatus rectum is x=aRequired volume isV=π∫
0
a
4 axdx
V=4a π [ x22 ]a0V=2π a3 cu . units
103 (a)
Volume=π∫0
4
16 xdx
Volume=8 π [ x2 ] 40
Volume=128π cu .units
104 (d)
Volume=π∫0
2
16 xdx
Volume=8 π [ x2 ] 20
Volume=32π cu .units
105 (a)
Volume=π∫0
4
(4 x−x2)dx
Volume=π {2 x2− x3
3 }40Volume=π (32−643 )Volume=32 π
3cu .units
106 (c)
Volume=π∫0
1
(4 x−4 x2)dx
Volume=4 π {x22 − x3
3 }10Volume=2 π
3cu .units
107 (b)
P a g e |42
Volume=π∫0
3
(3 x−x2)dx
Volume=π {3x22 − x3
3 }30Volume=π ( 273 −27
3 )Volume=9π
2cu .units
108 (d)
Volume=π∫0
1
(x2−x4 )dx
Volume=π { x33 − x5
5 }10Volume=π ( 215 )Volume=2 π
15cu .units
109 (c)
Volume=π∫0
2
(4 x2−x4 )dx
Volume=π {43 x3− x5
5 }20Volume=π ( 323 −32
5 )Volume=64π
15cu .units
110 (a)The area bounded by the curves as shown in fig.Curves is x2=4 yx=−3 , x=4Required volume isV=π∫
−3
4
y2dx
V=π∫−3
4
( x416 )dxV=
π16 [ x55 ] 4−3
V= π80
(1240+243 )
V=1267 π80
cu .units
111 (d)Equation of chord AB is y=x+3
Volume=π∫1
9
[− (x+3 )2+16 x ]dx
Volume=π [−( ( x+3 )3
3 )+8 x2]91Volume=π [ (−576+648 )−(−643 +8)]Volume=256 π
3cu .units
P a g e |43
112 (a)The area bounded by the curves as shown in fig.Curves is x2=4 y , y2=4 xSolving themA≡(4,4)Required volume isV=π∫
0
4
(4 x− x4
16 )dxV=π [2 x2− x5
80 ] 40V=π (32−102480 )V=32π ( 5−25 )V=96 π
5cu .units
P a g e |44
113 (d)
Volume=π∫0
1
(x−x4 )dx
Volume=π ( x22 − x5
5 )10Volume=3 π
15cu .units
114 (d)The area bounded by the curves as shown in fig.Curves isx2+ y2=25 and 3 x=4 ySolving themx=4 , y=3A≡ (4,4 ) ,B≡ (4,0 ) ,C≡(5,0)Required volume isV=π∫
0
4
y line2 dx+π∫
4
5
y¿̊2dx
¿
V=π∫0
4 9 x2
16dx+π∫
4
5
(25−x2)dx
V=9 π16 [ x33 ] 4
0+π [25 x− x3
3 ]54V=3 π
16(64 )+π (125−1253 −100+ 64
3 )V=36 π+14 π
3
V=50 π3
cu . units
115 (d)The area bounded by the curves as shown in fig.Curves isx2+ y2=36 and x+ y=6Solving themA≡ (6,0 ) , B≡ (0,6 )Required volume isV=Volume of sector OAB−Volume of ∆OAB
V=π∫0
6
(36−x2) dx−π∫0
6
(6−x )2dx
V=π∫0
6
(36−x2−36+12 x−x2)
V=π [6 x2− 2x33 ]60V=π (216−23 (216 ))V=72 π cu .units
116 (d)Curves isx2+ y2=36 and x=0 , x=3Required volume isV=π∫
0
3
y2
V=π∫0
3
(25−x2)dx
V=π [25 x− x3
3 ]30V=π (75−9)V=66π cu .units
117 (c)The area bounded by the curves as shown in fig.Curves isx2+ y2=4 and y2=3 xSolving them
P a g e |45
x=1 , y=√3A≡ (1 ,√3 ) ,B≡ (1,0 ) ,C≡(2,0)Required volume isV=π∫
0
1
y parabola2 dx+π∫
1
2
y¿̊2dx
¿
V=π∫0
1
3 xdx+π∫1
2
(4−x2)dx
V=π [3 x22 ] 10+π [4 x− x3
3 ]21V=π ( 32 +8−8
3−4+1
3 )V=π (4+ 32−73 )V=π
V=19 π6
cu .units
118 (d)The area bounded by the curves as shown in fig.Curves isx2
a2+ y
2
b2=1
Solving themA≡ (a ,0 ) , A ' ≡ (−a ,0 )Required volume isV=π∫
−a
a
y2dx
V=π∫−a
a
b2(1− x2
a2 )dxV= 2π b
2
a2∫0
a
(a2−x2)dx
V=2π b2
a2 [a2 x− x3
3 ]a0
V=2π b2
a2 (a3−a3
3 )V= 4π b
2a3
cu .units
119 (d)
Volume=π∫−4
4
9 (1− x2
16 )dxVolume=9π
16 ∫−44
(16−x2)dx
Volume=9π16 {16 x− x3
3 } 4−4Volume=9π
16 {(64−643 )−(−64+ 643 )}Volume=9π
16 (128−1283 )Volume=48 π cu .units
120 (d)
Volume=π∫−4
4
9 (1− x2
16 )dxVolume=18 π
16 ∫0
4
(16−x2)dx
Volume=9π8 {16 x− x3
3 }40Volume=9π
8 (64−643 )Volume=144 π
3cu .units
P a g e |46
121 (d)
Volume=π∫−3
3 49
(9− y2)dy
Volume=8π9 ∫
0
3
(9− y2)dy
Volume=8π9 (9 y− y3
3 )30Volume=24 π ( 23 )Volume=16π cu .units
122 (a)The area bounded by the curves as shown in fig.Curves isx2
a2+ y
2
b2=1
Major axis is AA’And Minor axis is BB’Solving themA≡ (a ,0 ) , B≡ (0 , b )Equation of chordxa+ yb=1
Required volume isV=π∫
0
a
¿¿¿)dxV=π∫
0
a
( b2a2 (a2−b2 )−b2
a2(a−x )2)
V= π b2
a2∫0
a
(a2−x2−a2+2ax−x2 )dx
V=π b2
a2 [ax2−2 x33 ]a0V= π b2
a2 (a3−2a33 )V= πab2
3cu . units
123 (c)The area bounded by the curves as shown in fig.Curves isx2− y2=a2
A≡ (a ,0 ) , B≡ (2a ,√3 a ) ,C≡(2a ,0)Required volume isV=π∫
a
2a
( x2−a2)dx
V=π [ x33 −a2x ]2aaV=π [8a33 −2a3−a3
3+a3]
V= 4π a3
3cu .units
124 (c)A sphere is generated by revolving the area ofSemi circle x2+ y2=r2 from x=−r ¿ x=rAs shown in figureRequired volume isP a g e |47
V=π∫−r
r
y2dx
V=π∫−r
r
(r 2−x2)dx
V=2π∫0
r
(r2−x2)dx
V=2π [r2 x− x3
3 ]r0V=2π (r3− r3
3 )V= 4π r
3
3cu .units
125 (a)
Volume=π∫−4
4
(16−x2)dx
Volume=2π∫0
4
(16−x2)dx
Volume=2π [16 x− x3
3 ]40Volume=2π (64−643 )Volume=2π ( 23 )Volume=256 π
3cu .units
126 (c)The area bounde by the curve is as shown
in figureA≡ (0 , r ) ,B≡ (h , r ) ,C≡(h ,0)Equation of AB is y=rRequired volume isV=π∫
0
h
y2dx
V=π∫0
h
r2dx
V=π r2 [x ]h0
V=π r2hcu .units
127 (d)The area bounde by the curve is as shown in figureA≡ (h , r ) ,B≡ (h ,0 ) ,Equation of OA isy= rx
hRequired volume isV=π∫
0
h r2x2
h2dx
V=π r2
h2 [ x33 ]hoV= π r2h
3cu .units
128 (d)
Area=∫−π
π
sin xdx
P a g e |48
Area=2∫0
π
sin x dx
Area=2 (−cos x )π0
Area=2(1+1)Area=4 sq .units
129 (b)
Area=∫0
4
y2dy
Area=13[ y3]4
0
Area=643sq .units
130 (a)
A=∫0
1
x2dx
A=13sq .units
131 (d)It is a square of diagoanal of length 4 units
A=(2√2 )2
A=8 sq .units
132 (a)Required area isA=∫
1
2 1xdx
A=[ log x ]21
A=( log2)sq .units
133 (d)Required area isA=∫
0
2
(2 x−x2)dx
A=[ x2− x3
3 ]20A=4−8
3
A=43sq .units
P a g e |49
134 (b)
∫0
16 /m2
(√16 x−mx )dx=23
x=16m2
[4× 23 x32−m x2
2 ]16/m2
0=23
¿> 83× 64m2
−m2256m4 =2
3¿>m=4
135 (d)Required area isA=∫
0
1
(√ x−x2)dx
A=[ 2x 323 − x3
3 ]10A=( 23−13 )A=1
3sq .units
P a g e |50
136 (b)Required area isA=πab
4−Areaof ∆OAB
A=5 (π−2 ) sq .units
137 (b)Required area isA=4 [ 12×1×1]A=2 sq .units
138 (b)Required area isA=∫
0
π /4
(cos x+sin x)dx
A=[sin x+cos x ] π /40
A=(√2−1 ) sq .units
139 (a)Required area isA=1
2(PQ )(AL)
A=12
(1 )(1)
A=12
[∴BC=2 , PQ=12BC=1 ,LA=1]
140 (a)The curve y=x3∧ y=√ x… ( i )( y ≥0)Points of intersection of curve (i) and y=x3…(ii)Are (0,0)(1,1)Required area isA=∫
0
1
( y1− y2)dx
A=∫0
1
(√ x¿−x3)dx¿
A=( 2 x32
3− x4
4 )10A=2
3− 14
A=8−312
A= 512
141 (a)Given function is y=|x−1|i.e.y=x−1x>1¿−x+1 x<1Required area is
A=12×1×1
P a g e |51
A=1
142 (b)Required area isA=∫
0
3
xdy
A=∫0
3 y2
4dy
A=[ y312 ]30A=27
12
A=94sq .units
143 (c)
Areaof ∆ AOB=12×1× 1
2
Areaof ∆ AOB=14
A=14sq .units
144 (c)
Areaofrhombus=12× (Product of diagonals )
Areaofrhombus=122ca2cb
Areaofrhombus=2c2
absq .units
145 (d)End points of latus rectum in 1st quadrant is (ae , b2a )Equation of tangent at (ae , b2a ) i.e. (3e , 253 )is
It intersects X-axis at ( 3e ,0) and y-axis at (0,3)
Areaof ∆OAB=12 ( 3e ).3= 9
2e
So, Area of quadrilateral ABCD=4. 92eAlso,b2=a2(1−e2)e2=1−b2
a2=1−5
9=49
¿>e=23
146 (c)We have,y=log x , x=1 , x=2Required area isA=∫
1
2
ydx
A=∫1
2
log xdx
P a g e |52
A=[x log x−x ]21
A=2 log2−2− log1+1A=2 log2−1A=2 log2−log eA=log 4− loge
A=log 4esq .units
147 (d)Required area isA=A (∆OAB )+A (∆OCD)
A=12×1×1+1
2×2×2
A=52sq .units
148 (a)Equation of parabola isy=x2…(i)And equation of the straight line isy=x… ( ii )
From (i) and (ii) ,we getx2−x=0¿>x=0∨x=1¿> y=0 or y=1Hence,the co-ordinates of their points of intersction are O(0,0) and P(1,1)∴ Required area between parabola and straight lineA=∫
0
1
xdx−∫0
1
x2dx
A=[ x22 −x3
3 ]10
A=[ 12− 13 ]A=1
6sq .units
149 (a)Required area isA=A (∆ ABD )+A(∆ ACE )
A=12×1×1+1
2×1×1
A=12+12
A=1 sq .units150 (a)Required area is
A=∫0
9
√x dx−∫3
9
( x−32 )dx
A=[ x32
32 ] 9
0−12[ x22 −3 x ]93
A=( 23 ,27)−12 {( 812 −27)−( 92−9)}A=18−9A=9 sq .units
151 (a)Required area isA= ∫
−π /2
π /2
ydx
A=2∫0
π /2
sin xdx
P a g e |53
A=2 [−cos x ]π /20
A=2 sq .units
152 (c)
x2
a2+ y
2
b2=1 , a>b ,
About major axisy2=b2(1− x2
a2 )y2=b2
a2(a2−x2 )
Volume=2π∫0
a
y2dx
Volume=2[π∫0a b2
a2(a2−x2 )dx ]
Volume=2 π b2
a2 (a2 x− x3
3 )a0Volume=2 π b
2
a2 (a3−a3
3 )Volume=2 π b
2
a2× 2a
3
3
Volume=43π ab2
153 (b)Required volume isV=π∫
0
2
x2dy
V=π∫0
2 y4dy
V=π4 [ y22 ] 20
V= π8
(16−0)
V=2π cu .units154 (b)Required volume is
V=π∫0
16
x2dy
V=π∫0
16 y4dy
V= π8
[ y2 ] 160
V=256 π8
V=32π cu .units155 (c)
the parabola x2=4 yi.e. y= 14 x2 passes throughthe points (0,0)(4,4) and (-4,4) and its axis of symmetry is X-axis.Now the volume generated by revolving the area bounded by the curvey= 14 x2,Y-axis y=0 and the line x=−3 , x=4 about X-axisV=π∫
−3
4
( 14 x2)2
dx
V= π16∫−3
4
x4dx
V=π16 [ x55 ] 4−3
V= π80 [ (4 )5−(−3 )5 ]
V= π80
[1024+243 ]
V= π80
[1267]
V=15.837 π c .u .156 (b)The volume of the solid generated by revolving the region bounded by the curve x= 2y ,Y-axis x=0The line y=1∧ y=4 about Y-axis is
P a g e |54
V=π∫1
4
[ f (x ) ]dy
V=π∫1
4
( 2y )2
dy
V=4 π∫1
4 dyy2
V=4 π [−1y ]41V=4 π [−14 +1]V=3π cu . units
157 (a)
Volumeof cone=13π ×16×4
Volumeof cone=64 π3
158 (b)
Required area isA=∫
−1
1
xdy
A=∫−1
1
y3dy
A=2∫0
1
y3dy
A=( 2 y44 )10A=1
2159 (a)The triangle area is
A=∫0
2
xdy
A=∫0
2 12ydy
A=12 [ y22 ]20
A=14[4−0 ]
A=1 sq .units
160 (c)Required area isA=∫
1
4
y . dx
A=∫1
4
4 √x .dx
P a g e |55
A=4 [ x32
32 ]41
A=83[4
32−13 /2]
A=83[8−1]
A=83×7
A=563sq .units
161 (b)Let LL’ be the latus rectum and S(1,0) be the focusOf the parabola y2=4ax∴Eqof letusrectum is x=1∴Required area=2× Areaof regionOSLO
A=2∫0
1
ydx
A=2∫0
1
2√ xdx
A=4∫0
1
x32 dx
A=4.[ x32
32 ]10
A=83
[132−0]A=8
3sq .units
162 (a)Required area=Area of regionOABO+¿Area of region BCDBA=∫
0
π
sin xdx+|∫π
2π
sin xdx|
A=[−cos x ]π
0+|[−cos x ]2ππ |
A=−cos π+cos0+|−cos2 π+cosπ|A=−(−1 )+1+|−1−1|A=2+2A=4 sq. units
163 (c)Equation of parabola is y=x2+2The line y=x passes through origin
Required area =Area under the parabola – Area under the lineA=∫
0
3
(x2+2 )dx−∫0
3
xdx
A=[ x33 +2x ] 3
0−( x2
3 )30
A=(9+6 )−( 92 )A=15−9
5
A=212
A=10.5 sq .units164 (d)The given curves are y2=x+1∧¿
P a g e |56
y2=−x+1These are two parabola’s whose vertices are (-1,0)And (1,0) respectively
Required Area isA=2[∫
−1
0
√x+1dx+∫0
1
√1−xdx ]
A=2{[ ( x+1 )32
32 ]
0
−1+[ (1−x )32
32 ]10}
A=43 { [1−0 ]+ [0+1 ] }
A=43×2
A=83sq .units
165 (b)The area bounded by the parabolax=f ( y )=1
2 √ y
Required Area isA=1
2∫14
xdy
A=12∫14
√ y dy
A=12∫14
y12 dy
A=12 [ y
32
32 ]41
A=13
[4 32−132 ]A= 4
3[8−1]
A=73sq .units
166 (c)Required Area isA=∫
a
b
log x .dx
A=[x . log x ]b
a−∫a
b
x . 1x dx
A=[x . log x−x ]ba
A=(b . log b−b ) .∫a
b
x . 1xdx
A=b ( log b−1 )−a (log a−1)A=b (log b−log e )−a (log a−log e)
A=−a log( be )+b log( ae )167 (a)The required area is shown by shaded portion
Required Area = 2× Area of region OABA=2×∫
0
π2
ydx
A=2∫0
π2
cos xdx
A=2 [sin x ]π20
P a g e |57
A=2(sin π2−sin 0)A=2(1−0)A=2 sq .units
168 (a)The area enclosed by the curves is shown in shaded region
At the point of intersection ofx2+ y2=32∧ y=x ,wehavex2+ y2=32=¿2 x2=32=¿ x2=16x=4Required Area =Areaof ∆OPQ+Area of ∆ PQA
A=∫0
4
xdx+∫4
4√2
√18−x2dx
A=[ x22 ]4
0+[ x2 √32−x2+ 322sin−1( x
4√2 )]4 √24
A=8+[0+16sin−11−8−16sin−1 1√2 ]A=16( π2 )−16 ( π4 )A=8π−4 πA=4 π sq .units
P a g e |58
169 (c)Equation of parabola is y2=16 x∴Focus=(4,0 )∧L .R . (¿' ) is x=4∴Required volumeis
V=∫0
4
π . y2dx
V=∫0
4
π .16 x .dx
V=16∫0
4
xdx
V=16 π [ x22 ]40V=16 π [8−0 ]V=128 π sq .units
170 (a)The given equation hyperbola isx2− y2=9=¿a2=9=¿a=3
Let S and S’ be its foci, A and A’ are the vertices of hyperbola the∴Required volume (V )=2×volume obtainedby revolving aboutX−axis the area ALS
∴Eccientricity of the hyperbola=√a2+b2a
=√2
∴Focus S=(3√2 ,0)
V=2π∫3
3√2
y2dx
V=2π∫3
3√2
(x2−9)dx
V=2π [ x33 −9 x ]3√23V=2π [36 √2
2−27√2−9+27]
V=2π [18√2−27√2+18 ]V=2π [18−9√2 ]V=18 π (2−√2 ) c .u
171 (b)The straight line y=2x passes through the originAnd P≡(0,6) Volume of the required region
V=π∫0
3
(2 x )2dx
V=4 π∫0
3
x2dx
V=4 π [ x33 ]30V= 4π
3[33−0]
V=36π cu .units172 (c)The curvey2=4axis symmetric about X-axis .Its vertex at origin and focus S≡ (a ,0 )Ends of latus rectum are (a ,2a )∧(a ,−2a )Required volume is
V=2π∫0
a
y2dx
V=2π∫0
a
4 axdx
V=8πa [ x22 ]a0V=8πa [ a22 −0 ]V=4 π a3 cub .units
173 (b)The lines y=2x+1 , y=3x+1 intersects at A(0,1).The linesy=3 x+1∧x=4 intersects P a g e |59
at B(4,13).The lines y=2x+1and x=4 intersects at C(4,9)
∴Therequired areaisA=Area (OABCD )−Area (OACD )
A=∫0
4
[ f ( x )−g( x)]dx
Where f (x )=3x+1 , g ( x )=2 x+1
A=∫0
4
[ (3 x+1 )−(2 x+1)]dx
A=∫0
4
xdx
A=[ x22 ]40A=16
2A=8 sq .units
174 (c)The sphere is the solid of revolution generated by the revolution of semi-circular area about its diameter
Equation of circle x2+ y2=a for the semi circle about X-axis the variable varies from x=−a ,x=aVolumeof sphere is
V=π∫−a
a
y2dx
V=π∫−a
a
(a2−x2 )dx
V=2π∫0
a
(a2−x2)dx
V=2π [a2 x− x3
3 ]a0V=2π [a3− a3
3 ]V= 4
3π a3cu .units
175 (b)
The curve is y=xRequired area = Area OCD + Area OABA=∫
−1
0
− ydx+∫0
2
ydx
A=∫−1
0
−xdx+∫0
2
xdx
A=−12
[x2 ]0
−1+12
[x2 ] 20
A=−12
(0−1 )+ 12(4−0)
A=52
176 (a)
The curve is x2a2
+ y2
b2=1=¿ y=b
a √a2−x2
Required area A = 4.Area OABOA=4∫
0
π /2 ba √a2−x2dx
Put x=a sinθ=¿dx=acosθ .dθ
P a g e |60
A=4∫0
π /2
a .b cosθ
A=4 a .b 12. π2
A=πab
177 (d)
The required volume isV=π∫
0
2
y2dx
V=π∫0
2
( x+1 )2dx
V=π∫0
2
(x2+2 x+1)dx
V=π [ x33 +x2+ x]20V=26 π
3
P a g e |61
178 (c)The equation of circle x=r .cosθ , y=r . sinθ∴ Required surface areaS=∫
0
π
2πy √( dxdθ )2
+( dydθ )2
dθ
S=∫r
π
2πr sinθ √r2+sin 2θ+r2+cos2θ dθ
S=2π r2∫0
π
sin θ .dθ
S=2π r2 (−cosθ )π1
S=2π r2(1+1)S=4 π r2
179 (c)
Volume of revolution isV=∫
2
4
π y2dx
V=∫2
4
π x4dx
V=[ π x55 ]42V= π
5(45−25 )
V= π5
(1024−32)
V=9923
π cu .units
180 (c)
We have,y2=4 x vertex =(0,0)The equation of latus rectum isGiven by x=a i.e.,x=1V=∫
0
1
π y2dx
V=∫0
1
π .4 x .dx
V=4 x [ x22 ]10V=2π
181 (c)
Area=∫1
e
log xdx
Area=∫1
e
1. log xdx
Area=[x log x−x ] e1
Area=1 sq .units182 (c)
Area=∫1
4
2√x dx
Area=2( 23 )[ x32 ]41
Area=43(8−1)
Area=283sq .units
183 (c)
Area=∫0
1
[ ( x+1 )−(x2+1 ) ]dx
Area=∫0
1
(x−x2)dx
Area=[ x22 −x3
3 ]10Area=1
2−13
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Area=16sq .units
184 (b)
Area=2∫0
a
2√a√ xdx
Area=4 √a ( 23 )[x32 ]a0
Area=83 √a(a√a)
Area=8 a2
3sq .units
185 (c)
Area=∫0
4
[ (3x+1 )− (2 x+1 ) ] dx
Area=[ x22 ]40Area=16
2Area=8 sq .units
186 (d)
Area=∫0
3
(x2+3)dx
Area=[ x33 +3 x]30Area=( 273 +9)−0Area=18 sq .units
187 (a)
Area=|∫−1
0
xdx|+∫02
xdx
Area=|[ x22 ] 0−1|+( x22 )20Area=1
2+2
Area=52sq .units
188 (b)Required area =Area of ellipse in first quadrant –A(∆ AOB)
Area=∫0
a ba √a2−x2dx−1
2ab
Area=ba {x2 √a2−x2+ a
2sin−1( xa )} a
0−12ab
Area=ba ( a22 . π2 )−12 ab
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Area=14πab−1
2ab
Area=ab(π−2)4
sq .units
189 (a)Required area isArea= ∫
0
4a /m2
2√a−x2dx− ∫0
4a /m2
mxdx
Area=2√a ( 23 )[ x32 ]4a /m2
0−m2
[ x2 ]4 a/m2
0
Area=4√a3 ( 8a√a
m3 )−m2 ( 16a2m4 )
Area= 8a2
3m3 sq .units
190 (d)Given curve isy2=4axAt t=1 , x=at=2 , x=4 a
Area=2∫a
4a
2√a√x dx
Area=4 √a ( 23 )[x32 ]4aa
Area=83 √a (8a√a−a√a )
Area=56 a2
3sq .units
191 (a)
Area=∫π /4
π /2
sin2 xdx+ ∫π /2
3π /4
sin 2 xdx
Area=[−cos2 x2 ]π /2
π /4+[−cos2x2 ]3π /4π /2
Area=−12
(−1−1)
Area=1 sq .units
192 (b)
Area=∫0
c
√kcdx
Area=2√k3
[ (√x )3 ]c0
Area=2√kc3
(c)
Area=2c3 √kc sq .units
193 (c)
Area=∫1
3
x3dx
Area=14
[x 4 ]31
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Area=14(80)
Area=20 sq .units
194 (a)
Area=∫0
π /6
cos3 xdx
Area=13
[sin3 x ] π /60
Area=13(1)
Area=13sq .units
195 (c)
Area=4∫0
r
√r2− x2dx
Area=4 [ x2 √r2−x2+ r2
2sin−1( xr )] r0
Area=4 ( r22 )( π2 )Area=π r2 sq .units
196 (c)
Volume=π∫0
2
x2dy
Volume=π∫0
2
y2dy
Volume=π3
[ y3 ]20
Volume=8π3cu .units
197 (c)
Volume=π∫0
4
y2dx
Volume=π∫0
4
r 2dx
Volume=πr 2 ( x ) 40
Volume=πr 2hcu .units
198 (d)
Volume=π∫0
h r2 x2
h2dx
Volume=π r2
3h2(x3)h
0
Volume=π r2h3
cu .unitsl
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199 (d)
Volume=π∫0
3
4 x2dx
Volume=4 π3
(x3)30
Volume=36π cu .units
P a g e |66
200 (a)
Volume=π∫−3
4 x4
16dx
Volume= π80
[ x5 ] 4−3
Volume= π80
(1024+243)
Volume=1267 π80
cu .units
201 (d)
Volume=π∫0
3
(25−x2)dx
Volume=π3
(75x−x3 )30
Volume=π3
(225−27)
Volume=66 π cu .units
202 (c)
Volume=π∫a
2a
( x2−a2 )dx
Volume=π [ x33 −a2 x ]2aaVolume=π ( 2a33 + 2a
3
3 )Volume=4 π a
3
3cu .units
203 (d)
Volume=π∫0
π2
sin 2 xdx
Volume=π2 {∫
0
π2
1.dx−∫0
π2
cos2 xdx}Volume=π
2 {[ x ]
π2
0−[ sin 2 x2 ]π20 }
Volume=π2 {π2−0}
Volume=π2
4sq .units
204 (c)
Volume=π∫0
1
3 x .dx+π∫1
2
(4−x2)dx
Volume=32π [ x2 ] 1
0+π [4 x− x3
3 ]21Volume=π {32+8−83−4+ 13 }Volume=19 π
6cu .units
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205 (d)
Volume=π∫1
4 1x2dx
Volume=π [−1x ]41
Volume=π (−14 +1)Volume=3 π
4cu .units
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