Chapter 6

Applications of Integration

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Problem 1

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = x + 1 $ , $ y = 0 $ , $ x = 0 $ , $ x = 2 $ ; about the x-axis

Rk
Rahat K.
Numerade Educator

Problem 2

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = \frac{1}{x} $ , $ y = 0 $ , $ x = 1 $ , $ x = 4 $ ; about the x-axis

Madi S.
Numerade Educator

Problem 3

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = \sqrt{x - 1} $ , $ y = 0 $ , $ x = 5 $ ; about the x-axis

RG
Raymond G.
Numerade Educator

Problem 4

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = e^x $ , $ y = 0 $ , $ x = -1 $ , $ x = 1 $ ; about the x-axis

RG
Raymond G.
Numerade Educator

Problem 5

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ x = 2 \sqrt{y} $ , $ x = 0 $ , $ y = 9 $ ; about the y-axis

RG
Raymond G.
Numerade Educator

Problem 6

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ 2x = y^2 $ , $ x = 0 $ , $ y = 4 $ ; about the y-axis

RG
Raymond G.
Numerade Educator

Problem 7

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = x^3 $ , $ y = x $ , $ x \ge 0 $ ; about the x-axis

RG
Raymond G.
Numerade Educator

Problem 8

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = 6 - x^2 $ , $ y = 2 $ ; about the x-axis

RG
Raymond G.
Numerade Educator

Problem 9

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y^2 = x $ , $ x = 2y $ ; about the y-axis

RG
Raymond G.
Numerade Educator

Problem 10

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ x = -2 - y^2 $ , $ x = y^4 $ ; about the y-axis

RG
Raymond G.
Numerade Educator

Problem 11

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = x^2 $ , $ x = y^2 $ ; about $ y = 1 $

RG
Raymond G.
Numerade Educator

Problem 12

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = x^3 $ , $ y = 1 $ , $ x = 2 $ ; about $ y = -3 $

RG
Raymond G.
Numerade Educator

Problem 13

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = 1 + \sec x $ , $ y = 3 $ ; about $ y = 1 $

RG
Raymond G.
Numerade Educator

Problem 14

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = \sin x $ , $ y = \cos x $ , $ 0 \le x \le \frac{\pi}{4} $ ; about $ y = -1 $

RG
Raymond G.
Numerade Educator

Problem 15

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = x^3 $ , $ y = 0 $ , $ x = 1 $ ; about $ x = 2 $

RG
Raymond G.
Numerade Educator

Problem 16

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ xy = 1 $ , $ y = 0 $ , $ x = 1 $ , $ x = 2 $ ; about $ x = -1 $

RG
Raymond G.
Numerade Educator

Problem 17

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ x = y^2 $ , $ x = 1 - y^2 $ ; about $ x = 3 $

RG
Raymond G.
Numerade Educator

Problem 18

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.
$$
y=x, y=0, x=2, x=4 ; \quad \text { about } x=1
$$

Amrita B.
Numerade Educator

Problem 19

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

$ \Re_1 $ about $ OA $

Amrita B.
Numerade Educator

Problem 20

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

$ \Re_1 $ about $ OC $

Amrita B.
Numerade Educator

Problem 21

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

$ \Re_1 $ about $ AB $

Amrita B.
Numerade Educator

Problem 22

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

$ \Re_1 $ about $ BC $

Amrita B.
Numerade Educator

Problem 23

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

$ \Re_2 $ about $ OA $

Amrita B.
Numerade Educator

Problem 24

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

$ \Re_2 $ about $ OC $

Amrita B.
Numerade Educator

Problem 25

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

$ \Re_2 $ about $ AB $

Amrita B.
Numerade Educator

Problem 26

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

$ \Re_2 $ about $ BC $

Amrita B.
Numerade Educator

Problem 27

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

$ \Re_3 $ about $ OA $

Amrita B.
Numerade Educator

Problem 28

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

$ \Re_3 $ about $ OC $

Amrita B.
Numerade Educator

Problem 29

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

$ \Re_3 $ about $ AB $

Amrita B.
Numerade Educator

Problem 30

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

$ \Re_3 $ about $ BC $

Amrita B.
Numerade Educator

Problem 31

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ y = e^{-x^2} $ , $ y = 0 $ , $ x = -1 $ , $ x = 1 $

(a) About the x-axis
(b) About $ y = -1 $

WZ
Wen Z.
Numerade Educator

Problem 32

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ y = 0 $ , $ y = \cos^2 x $ , $ \frac{-\pi}{2} \le x \le \frac{\pi}{2} $

(a) About the x-axis
(b) About $ y = 1 $

WZ
Wen Z.
Numerade Educator

Problem 33

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ x^2 + 4y^2 = 4 $

(b) About $ y = 2 $
(b) About $ x = 2 $

WZ
Wen Z.
Numerade Educator

Problem 34

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ y =x^2 $ , $ x^2 + y^2 = 1 $ , $ y \ge 0 $

(a) About the x-axis
(b) About the y-axis

WZ
Wen Z.
Numerade Educator

Problem 35

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the x-axis the region bounded by these curves.

$ y = \ln (x^6 + 2) $ , $ y = \sqrt{3 - x^3} $

Amrita B.
Numerade Educator

Problem 36

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the x-axis the region bounded by these curves.

$ y = 1 + xe^{-x^3} $ , $ y = \arctan x^2 $

Amrita B.
Numerade Educator

Problem 37

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.

$ y = \sin^2 x $ , $ y = 0 $ , $ 0 \le x \le \pi $ ; about $ y = -1 $

Amrita B.
Numerade Educator

Problem 38

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.

$ y = x $ , $ y = xe^{1 - \frac{x}{2}} $ ; about $ y = 3 $

Amrita B.
Numerade Educator

Problem 39

Each integral represents the volume of a solid. Describe the solid.

$ \pi \displaystyle \int_{0}^\pi \sin x dx $

Amrita B.
Numerade Educator

Problem 40

Each integral represents the volume of a solid. Describe the solid.

$ \pi \displaystyle \int_{-1}^1 (1 -y^2)^2 dy $

Amrita B.
Numerade Educator

Problem 41

Each integral represents the volume of a solid. Describe the solid.

$ \pi \displaystyle \int_{0}^1 (y^4 - y^8) dy $

Amrita B.
Numerade Educator

Problem 42

Each integral represents the volume of a solid. Describe the solid.

$ \pi \displaystyle \int_{1}^4 [3^2 - (3 - \sqrt{x})^2] dx $

Amrita B.
Numerade Educator

Problem 43

A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise 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 centimeters, are 0, 18, 58, 79, 94, 106, 117, 128, 63, 39, and 0. Use the Midpoint Rule to estimate the volume of the liver.

Amrita B.
Numerade Educator

Problem 44

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 listed in the table. Use the Midpoint Rule with $ n = 5 $ to estimate the volume of the log.

Amrita B.
Numerade Educator

Problem 45

(a) If the region shown in the figure is rotated about the x-axis to form a solid, use the Midpoint Rule with $ n = 4 $ to estimate the volume of the solid.
(b) Estimate the volume if the region is rotated about the y-axis. Again use the Midpoint Rule with $ n = 4 $.

Amrita B.
Numerade Educator

Problem 46

(a) A model for the shape of the bird's egg is obtained by rotating about the x-axis the region under the graph of
$$ f(x) = (ax^3 + bx^2 + cx + d) \sqrt{1 - x^2} $$
Use $ CAS $ to find the volume of such an egg.
(b) For a red-throated loon, $ a = -0.06 $, $ b = 0.04 $, $ c = 0.1 $, and $ d = 0.54 $. Graph $ f $ and find the volume of an egg of this species.

Amrita B.
Numerade Educator

Problem 47

Find the volume of the described solid $ S $.
A right circular cone with height $ h $ and base radius $ r $.

Amrita B.
Numerade Educator

Problem 48

Find the volume of the described solid $ S $.
A frustum of a right circular cone with height $ h $, lower base radius $ R $, and top radius $ r $.

Amrita B.
Numerade Educator

Problem 49

Find the volume of the described solid $ S $.
A cap of a sphere with radius $ r $ and height $ h $.

Amrita B.
Numerade Educator

Problem 50

Find the volume of the described solid $ S $.
A frustum of a pyramid with square base of side $ b $, square top of side $ a $, and height $ h $

What happens if $ a = b $? What happens if $ a = 0 $?

Amrita B.
Numerade Educator

Problem 51

Find the volume of the described solid $ S $.
A pyramid with height $ h $ and rectangular base with dimensions $ b $ and $ 2b $.

Amrita B.
Numerade Educator

Problem 52

Find the volume of the described solid $ S $.
A pyramid with height $ h $ and base an equilateral triangle with side $ a $ (a tetrahedron).

Amrita B.
Numerade Educator

Problem 53

Find the volume of the described solid $ S $.
A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3 cm, 4 cm, and 5 cm.

Amrita B.
Numerade Educator

Problem 54

Find the volume of the described solid $ S $.
The base of $ S $ is a circular disk with radius $ r $. Parallel cross sections perpendicular to the base are squares.

Amrita B.
Numerade Educator

Problem 55

Find the volume of the described solid $ S $.
The base of $ S $ is an elliptical region with boundary curve $ 9x^2 + 4y^2 = 36 $. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.

Amrita B.
Numerade Educator

Problem 56

Find the volume of the described solid $ S $.
The base of $ S $ is the triangular region with vertices $ (0, 0) $, $ (1, 0) $, and $ (0, 1) $. Cross-sections perpendicular to the y-axis are equilateral triangles.

Amrita B.
Numerade Educator

Problem 57

Find the volume of the described solid $ S $.
The base of $ S $ is the same base as in Exercise 56, but cross-sections perpendicular to the x-axis are squares.

Amrita B.
Numerade Educator

Problem 58

Find the volume of the described solid $ S $.
The base of $ S $ is the region enclosed by the parabola $ y = 1 - x^2 $ and the x-axis. Cross-sections perpendicular to the y-axis are squares.

Amrita B.
Numerade Educator

Problem 59

Find the volume of the described solid $ S $.
The base of $ S $ is the same base as in Exercise 58, but cross-sections perpendicular to the x-axis are isosceles triangles with height equal to the base.

Amrita B.
Numerade Educator

Problem 60

Find the volume of the described solid $ S $.
The base of $ S $ is the region enclosed by $ y = 2 - x^2 $ and the x-axis. Cross-sections perpendicular to the y-axis are quarter-circles.

Amrita B.
Numerade Educator

Problem 61

Find the volume of the described solid $ S $.
The solid $ S $ is bounded by circles that are perpendicular to the x-axis, intersect the x-axis, and have centers on the parabola $ y = \frac{1}{2} (1 - x^2) $, $ -1 \le x \le 1 $.

Carson M.
Numerade Educator

Problem 62

The base of $ S $ is a circular disk with radius $ r $. Parallel cross-sections perpendicular to the base are isosceles triangles with height $ h $ and unequal side in the base.
(a) Set up an integral for the volume of $ S $.
(b) By interpreting the integral as an area, find the volume of $ S $.

WZ
Wen Z.
Numerade Educator

Problem 63

(a) Set up an integral for the volume of a solid torus (the donut-shaped solid shown in the figure) with radii $ r $ and $ R $.
(b) By interpreting the integral as an area, find the volume of the torus.

WZ
Wen Z.
Numerade Educator

Problem 64

Solve Example 9 taking cross-sections to be parallel to the line of intersection of the two planes.

Amrita B.
Numerade Educator

Problem 65

(a) Cavalieri's Principle states that if a family of parallel planes gives equal cross-section areas for two solids $ S_1 $ and $ S_2 $ then the volumes of $ S_1 $ and $ S_2 $ are equal. Prove this principle.
(b) Use Cavalieri's Principle to find the volume of the oblique cylinder shown in the figure.

WZ
Wen Z.
Numerade Educator

Problem 66

Find the volume common to two circular cylinders, each with radius $ r $, if the axes of the cylinders intersect at right angles.

Amrita B.
Numerade Educator

Problem 67

Find the volume common to two spheres, each with radius $ r $, if the center of each sphere lies on the surface of the other sphere.

WZ
Wen Z.
Numerade Educator

Problem 68

A bowl is shaped like a hemisphere with diameter 30 cm. A heavy ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of $ h $ centimeters. Find the volume of water in the bowl.

WZ
Wen Z.
Numerade Educator

Problem 69

A hole of radius $ r $ is bored through the middle of a cylinder of radius $ R > r $ at right angles to the axis of the cylinder. Set up, but do not evaluate, an integral for the volume cut out.

Amrita B.
Numerade Educator

Problem 70

A hole of radius $ r $ is bored through the center of a sphere of radius $ R > r $. Find the volume of the remaining portion of the sphere.

WZ
Wen Z.
Numerade Educator

Problem 71

Some of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding the
volumes of wine barrels. (In fact Kepler published a book Stereometria doliorum in 1615 devoted to methods for finding the volumes of barrels.) They often approximated the shape of the sides by parabolas.
(a) A barrel with height $ h $ and maximum radius $ R $ is constructed by rotating about the x-axis the parabola $ y = R - cx^2 $, $ \frac{-h}{2} \le x \le \frac{h}{2} $, where c is a positive constant. Show that the radius of each end of the barrel is $ r = R - d $, where $ d = \frac{ch^2}{4} $.
(b) Show that the volume enclosed by the barrel is
$$ V = \frac{1}{3} \pi h (2R^2 + r^2 - \frac{2}{5} d^2) $$

WZ
Wen Z.
Numerade Educator

Problem 72

Suppose that a region $ \Re $ has area $ A $ and lies above the x-axis. When $ \Re $ is rotated about the x-axis, it sweeps out a solid with volume $ V_1 $. When $ \Re $ is rotated about the line $ y = -k $ (where $ k $ is a positive number), it sweeps out a solid with volume $ V_2 $. Express $ V_2 $ in terms of $ V_1 $, $ k $, and $ A $.

WZ
Wen Z.
Numerade Educator