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Chapter 6

Applications of Integration

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Problem 1

Let $ S $ be the solid obtained by rotating the region shown in the figure about the y-axis. Explain why it is awkward to use slicing to find the volume $ V $ of $ S $. Sketch a typical approximating shell. What are its circumference and height? Use shells to find $ V $.

Amrita B.
Numerade Educator

Problem 2

Let $ S $ be the solid obtained by rotating the region shown in the figure about the y-axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of $ S $. Do you think this method is preferable to slicing? Explain.

Amrita B.
Numerade Educator

Problem 3

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$ y = \sqrt[3]{x} $ , $ y = 0 $ , $ x = 1 $

Amrita B.
Numerade Educator

Problem 4

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

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

Amrita B.
Numerade Educator

Problem 5

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

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

Leon D.
Numerade Educator

Problem 6

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$ y = 4x - x^2 $ , $ y = x $

Amrita B.
Numerade Educator

Problem 7

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$ y = x^2 $ , $ y = 6x - 2x^2 $

Amrita B.
Numerade Educator

Problem 8

Let $ V $ be the volume of the solid obtained by rotating about the y-axis the region bounded by $ y = \sqrt{x} $ and y = x^2 $. Find $ V $ both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method.

Amrita B.
Numerade Educator

Problem 9

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ xy = 1 $ , $ x = 0 $ , $ y = 1 $ , $ y = 3 $

Amrita B.
Numerade Educator

Problem 10

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ y = \sqrt{x} $ , $ x = 0 $ , $ y = 2 $

Amrita B.
Numerade Educator

Problem 11

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ y = x^{\frac{3}{2}} $ , $ y = 8 $ , $ x = 0 $

Amrita B.
Numerade Educator

Problem 12

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ x = -3y^2 + 12y - 9 $ , $ x = 0 $

Amrita B.
Numerade Educator

Problem 13

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ x = 1 + (y - 2)^2 $ , $ x = 2 $

Amrita B.
Numerade Educator

Problem 14

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

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

Amrita B.
Numerade Educator

Problem 15

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$ y = x^3 $ , $ y = 8 $ , $ x = 0 $ ; about $ x = 3 $

Amrita B.
Numerade Educator

Problem 16

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$ y = 4 - 2x $ , $ y = 0 $ , $ x = 0 $ ; about $ x = -1 $

Amrita B.
Numerade Educator

Problem 17

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$ y = 4x - x^2 $ , $ y = 3 $ ; about $ x = 1 $

Amrita B.
Numerade Educator

Problem 18

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$ y = \sqrt{x} $ , $ x = 2y $ ; about $ x = 5 $

Amrita B.
Numerade Educator

Problem 19

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$ x = 2y^2 $ , $ y \ge 0 $ , $ x = 2 $ ; about $ y = 2 $

Amrita B.
Numerade Educator

Problem 20

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

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

Amrita B.
Numerade Educator

Problem 21

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ y = xe^{-x} $ , $ y = 0 $ , $ x = 2 $ ; about the y-axis

Amrita B.
Numerade Educator

Problem 22

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ y = \tan x $ , $ y = 0 $ , $ x = \frac{\pi}{4} $ ; about $ x = \frac{\pi}{2} $

Amrita B.
Numerade Educator

Problem 23

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ y = \cos^4 x $ , $ y = -\cos^4 x $ , $ \frac{-\pi}{2} \le x \le \frac{\pi}{2} $ ; about $ x = \pi $

Amrita B.
Numerade Educator

Problem 24

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

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

Amrita B.
Numerade Educator

Problem 25

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ x = \sqrt{\sin y} $ , $ 0 \le y \le \pi $ , $ x = 0 $ ; about $ y = 4 $

Amrita B.
Numerade Educator

Problem 26

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ x^2 - y^2 = 7 $ , $ x = 4 $ ; about $ y = 5 $

Amrita B.
Numerade Educator

Problem 27

Use the Midpoint Rule with $ n = 5 $ to estimate the volume obtained by rotating about the y-axis the region under the curve $ y = \sqrt{1 + x^3} $ , $ 0 \le x \le 1 $.

Amrita B.
Numerade Educator

Problem 28

If the region shown in the figure is rotated about the y-axis to form a solid, use the Midpoint Rule with $ n = 5 $ to estimate the volume of the solid.

Jacquelyn T.
Numerade Educator

Problem 29

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

$ \displaystyle \int_{0}^3 2 \pi x^5 dx $

Amrita B.
Numerade Educator

Problem 30

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

$ \displaystyle \int_{1}^3 2 \pi y \ln y dy $

Umar Sohail Q.
Numerade Educator

Problem 31

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

$ 2 \pi \displaystyle \int_{1}^4 \frac{y + 2}{y^2} dy $

Amrita B.
Numerade Educator

Problem 32

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

$ \displaystyle \int_{0}^1 2 \pi (2 - x)(3^x - 2^x) dx $

Amrita B.
Numerade Educator

Problem 33

Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the y-axis the region enclosed by these curves.

$ y = x^2 - 2x $ , $ y = \frac{x}{x^2 + 1} $

Amrita B.
Numerade Educator

Problem 34

Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the y-axis the region enclosed by these curves.

$ y = e^{\sin x} $ , $ y = x^2 - 4x + 5 $

Amrita B.
Numerade Educator

Problem 35

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 = \sin^4 x $ , $ 0 \le x \le \pi $ ; about $ x = \frac{\pi}{2} $

Amrita B.
Numerade Educator

Problem 36

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^3 \sin x $ , $ y = 0 $ , $ 0 \le x \le \pi $ ; about $ x = -1 $

Amrita B.
Numerade Educator

Problem 37

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ y = -x^2 + 6x - 8 $ , $ y = 0 $ ; about the y-axis

Amrita B.
Numerade Educator

Problem 38

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ y = -x^2 + 6x - 8 $ , $ y = 0 $ ; about the x-axis

Amrita B.
Numerade Educator

Problem 39

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

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

WZ
Wen Z.
Numerade Educator

Problem 40

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

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

WZ
Wen Z.
Numerade Educator

Problem 41

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ x^2 + (y - 1)^2 = 1 $ ; about the y-axis

Amrita B.
Numerade Educator

Problem 42

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ x = (y - 3)^2 $ , $ x = 4 $ ; about $ y = 1 $

Amrita B.
Numerade Educator

Problem 43

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ x = (y - 1)^2 $ , $ x - y = 1 $ ; about $ x = -1 $

Amrita B.
Numerade Educator

Problem 44

Let $ T $ be the triangular region with vertices $ (0, 0) $, $ (1, 0) $, and $ (1, 2) $, and let $ V $ be the volume of the solid generated when $ T $ is rotated about the line $ x = a $, where $ a > 1 $. Express $ a $ in terms of $ V $.

Jacquelyn T.
Numerade Educator

Problem 45

Use cylindrical shells to find the volume of the solid.

$ A $ sphere of radius $ r $

Amrita B.
Numerade Educator

Problem 46

Use cylindrical shells to find the volume of the solid.

The solid torus of Exercise 6.2.63

Amrita B.
Numerade Educator

Problem 47

Use cylindrical shells to find the volume of the solid.

$ A $ right circular cone with height $ h $ and base radius $ r $

Amrita B.
Numerade Educator

Problem 48

Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height $ h $, as shown in the figure.
(a) Guess which ring has more wood in it.
(b) Check your guess: Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius $ r $ through the center of a sphere of radius $ R $ and express the answer in terms of $ h $.

Carson M.
Numerade Educator