Volumes by Slicing

7.3

Solids of Revolution

Find the volume of the solid generated by revolving the regionsabout the x-axis.2y x x and y 0 bounded by

12

0

r dx 1

22

0

dx xx

Find the volume of the solid generated by revolving the regionsabout the x-axis.2y 3x x and y 0 bounded by

02

3

dxr

20

2

3

dx3x x

Find the volume of the solid generated by revolving the regionsabout the y-axis.1

2y x, x 0, y 2 bounded by

22

0

r dy 2

2

0

d2y y

Find the volume of the solid generated by revolving the regionsabout the x-axis.2y 2sin2x, 0 x bounded by

/ 22

0

xr d

/ 2

2

0

dx2sin2x

Find the volume of the solid generated by revolving the regionsabout the line y = -1.2y 3 x , y 1 bounded by

22

2

dxr

2 22

2

3 x 1 dx

Let R be the first quadrant region enclosed by the graph ofxy 2e and x k

a) Find the area of R in terms of k.

b) Find the volume of the solid generated when R is rotated about the x-axis in terms of k.

c) What is the volume in part (b) as k approaches infinity?

NO CALCULATOR

Let R be the first quadrant region enclosed by the graph ofxy 2e and x k

a) Find the area of R in terms of k.k

x x k k0

0

2e dx 2e | 2e 2

Let R be the first quadrant region enclosed by the graph ofxy 2e and x k

b) Find the volume of the solid generated when R is rotated about the x-axis in terms of k.

k

2x

0

2e dx

k

2x

0

4 e dxu 2x

du 2dx

uk

0

e d2 u 2x k

02 e | 2k2 e 2

Let R be the first quadrant region enclosed by the graph ofxy 2e and x k

c) What is the volume in part (b) as k approaches infinity?

2k

klim 2 e 2 2

Let R be the region in the first quadrant under the graph of

3

8y for 1, 8x

a) Find the area of R.

b) The line x = k divides the region R into two regions. If the part of region R to the left of the line is 5/12 of the area of the whole region R, what is the value of k?

c) Find the volume of the solid whose base is the region R and whose cross sections cut by planes perpendicular to the x-axis are squares.

CALCULATOR REQUIRED

Let R be the region in the first quadrant under the graph of

3

8y for 1, 8x

a) Find the area of R.

8

31

8 dx 36x

Let R be the region in the first quadrant under the graph of

3

8y for 1, 8x

b) The line x = k divides the region R into two regions. If the part of region R to the left of the line is 5/12 of the area of the whole region R, what is the value of k?

k

31

8 dxx

A

5 3612

2/ 3 k112x | 15

2/32k 11 12 5

k 3.375

Let R be the region in the first quadrant under the graph of

3

8y for 1, 8x

c) Find the volume of the solid whose base is the region R and whose cross sections cut by planes perpendicular to the x-axis are squares.

Cross Sections

28

21

8 dx 192x

The base of a solid is the circle . Each section of thesolid cut by a plane perpendicular to the x-axis is a square withone edge in the base of the solid. Find the volume of thesolid in terms of a. (NO PI)

2 2 2x y a

a 2

2 2

a

2 a x dx

a

2 2

a

4a 4x dx

3

2 aa

4x4a x |3

316 a3

Let R be the region in the first quadrant that is enclosed by thegraph of f x ln x 1 , the x-axis and the line x = e. What is

the volume of the solid generated when R is rotated about theline y = -1?A.

5.037 B. 6.545 C. 10.073 D. 20.146 E. 28.686

CALCULATOR REQUIRED

e

2 2

0

ln x 1 1 1 dx 20.14627352 D

Let R be the region marked in the first quadrant enclosed bythe y-axis and the graphs of as shown in the figure below

2y 4 x and y 1 2sinx

R

a) Setup but do not evaluate the integral representing the volume of the solid generated when R is revolved around the x-axis.

b) Setup, but do not evaluate the integral representing the volume of the solid whose base is R and whose cross sections perpendicular to the x-axis are squares.

1.102

2 22

0

4 x 1 2sinx dx

1.102 22

0

4 x 1 2sinx dx

Let R be the region in the first quadrant bounded above by thegraph of f(x) = 3 cos x and below by the graph of 2xg x e

a) Setup, but do not evaluate, an integral expression in terms of a single variable for the volume of the solid generated when R is revolved about the x-axis.

b) Let the base of a solid be the region R. If all cross sections perpendicular to the x-axis are equilateral triangles, setup, but do not evaluate, an integral expression of a single variable for the volume of the solid.

20.836 22 x

0

3cosx e dx

20.836 2

x

0

3 3cosx e dx4

The volume of the solid generated by revolving the first quadrantregion bounded by the curve and the lines x = ln 3 andy = 1 about the x-axis is closest to

x / 2y e

a) 2.79 b) 2.82 c) 2.85 d) 2.88 e) 2.91

ln3

2 2x / 2

0

e 1 dx

The base of a solid is a right triangle whose perpendicular sideshave lengths 6 and 4. Each plane section of the solidperpendicular to the side of length 6 is a semicircle whosediameter lies in the plane of the triangle. The volume of the solidin cubic units is: a) 2pi b) 4pi c) 8pi d) 16pi e) 24pi

26

0

1 1 dx2

2 x2 3

2

322

0

222

0

24 2

0

2 4

The volume of the solid generated by rotating about the x-axis

the region enclosed between the curve y 3x and the line y 6x is given by

A. 6x 3x dx

B. 6x 3x dx

C. 9x 36x dx

D. 36x 9x d

2

0

22

0

x

E. 6x 3x dx

CALCULATOR REQUIRED

NO CALCULATORThe base of a solid is the region in the first quadrant bounded by

the curve y sinx for 0 x . If each cross section of the solid perpendicular to the x-axis is a square, the volume of thesolid, in cu

bic units, is:A. 0 B. 1 C. 2 D. 3 E. 4

2

00 0

sinx dx sinxdx cos x | 1 1 2 C

Let R be the region in the first quadrant above by the graph of f x 2Arc tanx and below by the graph of y = x. What is the volume

of the solid generated when R is rotated about the x-axis?A. 1.21

B. 2.28 C. 2.69 D. 6.66 E. 7.15

CALCULATOR REQUIRED

NO CALCULATORThe base of a solid is a right triangle whose perpendicular sideshave lengths 6 and 4. Each plane section of the solid perpendicularto the side of length 6 is a semicircle whose diameter lies in theplane of the triangle. The volume, in cubic units, of the solid is:A. 2 B. 4 C. 8 D. 16 E. 24

2

6 62

0 0

3 60

2 x1 13 dx x dx2 2 18

x |544 B

x / 2

The volume of the solid generated by revolving the first quadrant

region bounded by the curve y e and the lines x = ln 3 and y = 1about the x-axis is:A. 2.80 B. 2.83 C. 2.86

D. 2.89 E. 2.92

CALCULATOR REQUIRED

CALCULATOR REQUIRED

2 2

The region S is represented by the area between the graphs of

f x 0.5x 2x 4 and g x 2 4 4x x . Write, but do

not evaluate, a definite integral which represents:a. the volume of a solid with base S if eac

h cross section ofthe solid perpendicular to the x-axis is a semi-circle.

b. the volume generated by rotating region S around the line y = 5.

24

0

g x f xdx

2 2

4

2 2

0

5 f x 5 g x dx