r Exercises 5.2

Figure 530 (a)

E X A M P L E ' S The region in the first quadrant bounded by the graphs of y = i* and y = 2x is revolved about the y-axis. Find the volume of the resulting solid.

SOLUTION The region and a typical horizontal rectangle are shown in Figure 5.30(a). We wish to integrate with respect to y, so we solve the given equations for x in terms of y, obtaining

= 2y"\ b and

Figure 5.30(b) illustrates the volume generated by the region and the washer generated by the rectangle. We note the following:

thickness of washer: dy outer radius: 2y1'3

inner radius: | y volume: n[(2y1/3)2 - (±y)2]dy = n(4y2/3 - \y2) dy

Applying the limit of sums operator J0 gives us the volume:

f8

V= n{Ay2l3-\y2)dy. Jo

n [fy5'3 1 -.,31 — 512^. i2 y J0 - -w71 107.2

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EXERCISES 5.2

Exer. 1-4: Set up an integral that can be used to find the volume of the solid obtained by revolving the shaded region about the indicated axis.

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I IS II i! 4 •* I

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CHAPTER 5 Applications of the Definite Integral

ky

+ (3,4)

* - V25 - y2

+ ( 3 . - 4 )

y = -2X2. + 2

Exer. 5-24: Sketch the region R bounded by the graphs of the equations, and find the volume of the solid generated if R is revolved about the indicated axis. 5 y = l/x, x = 1, x = 3, y = 0; x-axis 6 y = V^c. x = 4, y = 0; x-axis

7 y = x2 — 4x, y = 0; x-axis 8 y = x3, x = - 2 , y = 0; x-axis a y = x2, y = 2; y-axis 0 y = l/x, y = l, y = 3, x = 0; y-axis 1 x = 4y — y2, x = 0; y-axis 2 y = x, y = 3, x = 0; y-axis 3 y = x2, y = 4 —x2; x-axis 4 x = y3, x2 + y = 0; x-axis 5 y = x , x + y = 4, x = 0; x-axis 6 y = (x - l)2 + 1, y = -(x - 1)2 + 3 ; j.axjs

7 y2 = x, 2y = x; y-axis

8 y = 2x, y = 4x2; y-axis

2 0 x + y = l, x —y = —1, x = 2 ; y-axis 21 y = sin2x, x = 0, x = n, y = 0; x-axis

(Hint: Use a half-angle formula.) 22 y = 1+cos3x, x = 0, x = 27T, y = 0; x-axis

(Hint: Use a half-angle formula.) 23 y = sinx, y = c o s x , x = 0, x = n/4; x-axis

(Hint: Use a double angle formula.) 24 y = secx, y = sinx, x = 0, x = TT/4; x-axis

Exer. 25-26: Sketch the region R bounded by the graphs of the equations, and find the volume of the solid gen-erated if R is revolved about the given line.

25 y = x2, y = 4 ( a ) y = 4 (b )y = 5 (c) x = 2 (d) x = 3

26 y = Vx. y = 0, x = 4 • (a) x = 4 (b) x = 6 (c) y = 2 (d) y = 4

Exer. 27-28: Set up an integral that can be used to find the volume of the solid generated by revolving the shaded region about the line (a) y = - 2 , (b) y = 5, (c) x = 7, and ( d ) x = - 4 .

27 ky

28

9 x = y2, x - y = 2; y-axis

430 CHAPTER 6 Applications of Integration

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lr

52. Volume of a Lab Glass A glass container can be modeled by revolving the graph of

y = VOlx* 2.95,

2.2x2 + 10.9* + 22.2, 0 < x < 11.5 11.5 < x < 15

about the x-axis, where x and y are measured in centimeters. Use a graphing utility to graph the function and find the volume of the container.

53. Find the volume of the solid generated if the .upper half of the ellipse 9x2 + 25y2 = 225 is revolved about (a) the x-axis to form a prolate spheroid (shaped like a football). (b) the y-axis to form an oblate spheroid (shaped like half of a

candy).

« 5 -

y A

5 -

-5 i -• Figure for 53(a) Figure for 53(b)

54. Minimum Volume The arc of y = 4 — (x2/4) on the interval [0, 4] is revolved about the line y — b (see figure). (a) Find the volume of the resulting solid as a function of b. (b) Use a graphing utility to graph the function in part (a), and

use the graph to approximate the value of b that minimizes the volume of the solid.

(c) Use calculus to find the value of b that minimizes the vol-ume of the solid, and compare the result with the answer to part (b).

y = b

Figure for 54 Figure for 56

IS 55. Water Depth in a Tank A tank on a water tower is a sphere of radius 50 feet. Determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity. (Note: Use the root-finding capabilities of a graphing utility after evaluating the definite integral.)

56. Modeling Data A draftsman is asked to determine the amount of material required to produce a machine part (see figure in first column). The diameters d of the part at equally spaced points x are listed in the table. The measurements are listed in centimeters.

X

d

X

d

0

4.2

6

5.8

1

3.8

7

5.4

2

4.2

8

4.9

3

4.7

9

4.4

4

5.2

10

4.6

5

5.7

(a) Use these data with Simpson's Rule to approximate the volume of the part.

(b) Use the regression capabilities of a graphing utility to find a fourth-degree polynomial through the points representing the radius of the solid. Plot the data and graph the model.

(c) Use a graphing utility to approximate the definite integral yielding the volume of the part. Compare the result with the answer to part (a).

57. Think About It Match each integral with the solid whose volume it represents, and give the dimensions of each solid. (a) Right circular cylinder (b) Ellipsoid

(d) Right circular cone (e) Torus (c) Sphere '•h

(i) m dx

(ii) 7T r2dx

ii) 771 (Vr2 - x2fdx (in

(iv) TT 1 - - I A

(v) TT| [{R + Vr2 - x2)2 -{R- Jr2-x2f]dx

58. Find the volume of concrete in a ramp that is 3 meters wide and whose cross sections are right triangles with base 10 meters and height 2 meters (see figure).

59. Find the volume of the solid whose base is bounded by the graphs of y = x + 1 and y = x2 - 1, with the indicated cross sections taken perpendicular to the x-axis. (a) Squares (b) Rectangles of height 1

SECTION 6.2 Volume: The Disk Method 431

^^ 60. Find the volume of the solid whose base is bounded by the

circle x2 + y2 = 4, with the indicated cross sections taken

r

perpendicular to the x-axis. (a) Squares (b) Equilateral triangles

2 y

(c) Semicircles (d) Isosceles right triangles

61. The base of a solid is bounded by v = x3, y — 0, and x — 1. Find the volume of the solid for each of the following cross sec-tions (taken perpendicular to the y-axis): (a) squares, (b) semi-circles, (c) equilateral triangles, and (d) semiellipses whose heights are twice the lengths of their bases.

62. Find the volume of the solid of intersection (the solid common to both) of the two right circular cylinders of radius r whose axes meet at right angles (see figure).

Two intersecting cylinders Solid of intersection

FOR FURTHER INFORMATION For more information on this problem, see the article "Estimating the Volumes of Solid Figures with Curved Surfaces" by Donald Cohen in Mathematics Teacher. To view this article, go to the website www.matharticles.com. 63. Cavalieri's Theorem Prove that if two solids have equal

altitudes and all plane sections parallel to their bases and at equal distances from their bases have equal areas, then the solids have the same volume (see figure).

64. A manufacturer drills a hole through the center of a metal sphere of radius R. The hole has a radius r. Find the volume of the resulting ring.

65. For the metal sphere in Exercise 64, let R = 5. What value of r will produce a ring whose volume is exactly half the volume of the sphere?

66. The solid shown in the figure has cross sections bounded by the graph of

\x\° + \y\° = 1

where 1 < a < 2. (a) Describe the cross section when a = 1 and a = 2. (b) Describe a procedure for approximating the volume of the

solid.

67. Two planes cut a right circular cylinder to form a wedge. One plane is perpendicular to the axis of the cylinder and the second makes an angle of 6 degrees with the first (see figure). (a) Find the volume of the wedge if 6 — 45°. (b) Find the volume of the wedge for an arbitrary angle 6.

Assuming that the cylinder has sufficient length, how does the volume of the wedge change as 6 increases from 0° to 90°?

c :m;: H,-R2*r

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Area of/?, = area of R2

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f**-SECTION 6.2 VOLUMES 391

6.2 Exercises 1-18 mi Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified fine. Sketch the region, the solid, and a typical disk or washer.

1. y = x2, x = 1, y = 0; about the x-axis

2. x + 2y = 2, x — 0 y = 0; about the x-axis

3. y = l/x, x — 1, x = 2, y = 0; about the x-axis

4. y = vJC—T, x = 2, x = 5, y = 0; aboutthe-x-axis

5. y = *2, 0 *£ x =s 2, y = 4, x — 0; about the y-axis

6. x = y - y2, x = 0; about the y-axis

7. y = x2, y2 = x; about the x-axis '

8. y = sec x, y = 1, x = —1, x = 1; about the x-axis

9. y2 = x, x = 2y; about the y-axis

0. y = x2 / \ x = 1, y = 0; about the y-axis = x2 / \ x =

1. y = x, y = V*; about y = 1

= x2, y = 4; about y = 4

- x4, y = 1; about y = 2

4. y = l/x2, y = 0, x = 1, x = 3; about y = - 1

5. x = y2, x = 1; about x = 1

6. y = x, y = Vx; about x = 2

'• y = x2, x = y2; about x — - 1

8. y = x, y = 0, x = 2, x = 4; about x = 1

27. 0t3 about OA

29. ^ 3 about AS

28. &3 about OC

30. Sft3 about SC

31-36 nil Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

31. y = tan3x, y = 1, x = 0; about y = 1

32. y = (x - 2)4, 8x - y = 16; about x = 10

33. y = 0, y = sin x, 0 «£ x «s ir, about y = 1

34. y.= 0, y = sinx, 0 ss x =s IT; about y = - 2

35. x2 - y2 = 1, x = 3; about x = - 2

36. 2x + 3y = 6, (y - I)2 = 4 - x; about x = - 5

37-38 nil Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approxi-mately) the volume of the solifl obtained by rotating about the x-axis the region bounded by these curves.

38. y = x4, y = 3x - x3 37. y = x2, y = y/x + 1

ICASl 39-40 llll Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

39. y = sin2x, y = 0, 0 =£ x =s IT; about y = - 1

40. y = x2 - 2x, y = x COS(TTX/4); about y = 2 H

19-30 mi Refer to the figure and find the volume generated by rotating the given region about the specified line.

9- % about OA

' ^ ^ bout AB

'• . about OA S- % about AB

20. % about OC

22. &, about BC

24. 9t2 about OC

26. a 2 about SC

41-44 llll Each integral represents the volume of a solid. Describe the solid.

41. 7t cos2xdx Jo

43. 7 rP (y 4 -y%)dy Jo

44. i r J ^ K l + c o s x ) 2 - l2]<zx

42. 'ify*'

45. A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ other-wise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centi-meters, are 6, 18, 58, 79, 94, 106, 117, 128, 63, 39, and 0. Use the Midpoint Rule to estimate the volume of the liver.

46. A log 10 m long is cut at 1-meter intervals and its cross-sectional areas A (at a distance x from the end of the log) are