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Calculus Volume 2

2.3 Volumes of Revolution: Cylindrical Shells

Calculus Volume 22.3 Volumes of Revolution: Cylindrical Shells
  1. Preface
  2. 1 Integration
    1. Introduction
    2. 1.1 Approximating Areas
    3. 1.2 The Definite Integral
    4. 1.3 The Fundamental Theorem of Calculus
    5. 1.4 Integration Formulas and the Net Change Theorem
    6. 1.5 Substitution
    7. 1.6 Integrals Involving Exponential and Logarithmic Functions
    8. 1.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  3. 2 Applications of Integration
    1. Introduction
    2. 2.1 Areas between Curves
    3. 2.2 Determining Volumes by Slicing
    4. 2.3 Volumes of Revolution: Cylindrical Shells
    5. 2.4 Arc Length of a Curve and Surface Area
    6. 2.5 Physical Applications
    7. 2.6 Moments and Centers of Mass
    8. 2.7 Integrals, Exponential Functions, and Logarithms
    9. 2.8 Exponential Growth and Decay
    10. 2.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  4. 3 Techniques of Integration
    1. Introduction
    2. 3.1 Integration by Parts
    3. 3.2 Trigonometric Integrals
    4. 3.3 Trigonometric Substitution
    5. 3.4 Partial Fractions
    6. 3.5 Other Strategies for Integration
    7. 3.6 Numerical Integration
    8. 3.7 Improper Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  5. 4 Introduction to Differential Equations
    1. Introduction
    2. 4.1 Basics of Differential Equations
    3. 4.2 Direction Fields and Numerical Methods
    4. 4.3 Separable Equations
    5. 4.4 The Logistic Equation
    6. 4.5 First-order Linear Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  6. 5 Sequences and Series
    1. Introduction
    2. 5.1 Sequences
    3. 5.2 Infinite Series
    4. 5.3 The Divergence and Integral Tests
    5. 5.4 Comparison Tests
    6. 5.5 Alternating Series
    7. 5.6 Ratio and Root Tests
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Chapter Review Exercises
  7. 6 Power Series
    1. Introduction
    2. 6.1 Power Series and Functions
    3. 6.2 Properties of Power Series
    4. 6.3 Taylor and Maclaurin Series
    5. 6.4 Working with Taylor Series
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  8. 7 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 7.1 Parametric Equations
    3. 7.2 Calculus of Parametric Curves
    4. 7.3 Polar Coordinates
    5. 7.4 Area and Arc Length in Polar Coordinates
    6. 7.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Learning Objectives

  • 2.3.1. Calculate the volume of a solid of revolution by using the method of cylindrical shells.
  • 2.3.2. Compare the different methods for calculating a volume of revolution.

In this section, we examine the method of cylindrical shells, the final method for finding the volume of a solid of revolution. We can use this method on the same kinds of solids as the disk method or the washer method; however, with the disk and washer methods, we integrate along the coordinate axis parallel to the axis of revolution. With the method of cylindrical shells, we integrate along the coordinate axis perpendicular to the axis of revolution. The ability to choose which variable of integration we want to use can be a significant advantage with more complicated functions. Also, the specific geometry of the solid sometimes makes the method of using cylindrical shells more appealing than using the washer method. In the last part of this section, we review all the methods for finding volume that we have studied and lay out some guidelines to help you determine which method to use in a given situation.

The Method of Cylindrical Shells

Again, we are working with a solid of revolution. As before, we define a region R,R, bounded above by the graph of a function y=f(x),y=f(x), below by the x-axis,x-axis, and on the left and right by the lines x=ax=a and x=b,x=b, respectively, as shown in Figure 2.25(a). We then revolve this region around the y-axis, as shown in Figure 2.25(b). Note that this is different from what we have done before. Previously, regions defined in terms of functions of xx were revolved around the x-axisx-axis or a line parallel to it.

This figure has two graphs. The first graph is labeled “a” and is an increasing curve in the first quadrant. The curve is labeled “y=f(x)”. The curve starts on the y-axis at y=a. Under the curve, above the x-axis is a shaded region labeled “R”. The shaded region is bounded on the right by the line x=b. The second graph is a three dimensional solid. It has been created by rotating the shaded region from “a” around the y-axis.
Figure 2.25 (a) A region bounded by the graph of a function of x.x. (b) The solid of revolution formed when the region is revolved around the y-axis.y-axis.

As we have done many times before, partition the interval [a,b][a,b] using a regular partition, P={x0,x1,…,xn}P={x0,x1,…,xn} and, for i=1,2,…,n,i=1,2,…,n, choose a point xi*[xi1,xi].xi*[xi1,xi]. Then, construct a rectangle over the interval [xi1,xi][xi1,xi] of height f(xi*)f(xi*) and width Δx.Δx. A representative rectangle is shown in Figure 2.26(a). When that rectangle is revolved around the y-axis, instead of a disk or a washer, we get a cylindrical shell, as shown in the following figure.

This figure has two images. The first is a cylindrical shell, hollow in the middle. It has a vertical axis in the center. There is also a curve that meets the top of the cylinder. The second image is a set of concentric cylinders, one inside of the other forming a nesting of cylinders.
Figure 2.26 (a) A representative rectangle. (b) When this rectangle is revolved around the y-axis,y-axis, the result is a cylindrical shell. (c) When we put all the shells together, we get an approximation of the original solid.

To calculate the volume of this shell, consider Figure 2.27.

This figure is a graph in the first quadrant. The curve is increasing and labeled “y=f(x)”. The curve starts on the y-axis at f(x*). Below the curve is a shaded rectangle. The rectangle starts on the x-axis. The width of the rectangle is delta x. The two sides of the rectangle are labeled “xsub(i-1)” and “xsubi”.
Figure 2.27 Calculating the volume of the shell.

The shell is a cylinder, so its volume is the cross-sectional area multiplied by the height of the cylinder. The cross-sections are annuli (ring-shaped regions—essentially, circles with a hole in the center), with outer radius xixi and inner radius xi1.xi1. Thus, the cross-sectional area is πxi2πxi12.πxi2πxi12. The height of the cylinder is f(xi*).f(xi*). Then the volume of the shell is

Vshell=f(xi*)(πxi2πxi12)=πf(xi*)(xi2xi12)=πf(xi*)(xi+xi1)(xixi1)=2πf(xi*)(xi+xi12)(xixi1).Vshell=f(xi*)(πxi2πxi12)=πf(xi*)(xi2xi12)=πf(xi*)(xi+xi1)(xixi1)=2πf(xi*)(xi+xi12)(xixi1).

Note that xixi1=Δx,xixi1=Δx, so we have

Vshell=2πf(xi*)(xi+xi12)Δx.Vshell=2πf(xi*)(xi+xi12)Δx.

Furthermore, xi+xi12xi+xi12 is both the midpoint of the interval [xi1,xi][xi1,xi] and the average radius of the shell, and we can approximate this by xi*.xi*. We then have

Vshell2πf(xi*)xi*Δx.Vshell2πf(xi*)xi*Δx.

Another way to think of this is to think of making a vertical cut in the shell and then opening it up to form a flat plate (Figure 2.28).

This figure has two images. The first is labeled “a” and is of a hollow cylinder around the y-axis. On the front of this cylinder is a vertical line labeled “cut line”. The height of the cylinder is “y=f(x)”. The second figure is labeled “b” and is a shaded rectangular block. The height of the rectangle is “f(x*), the width of the rectangle is “2pix*”, and the thickness of the rectangle is “delta x”.
Figure 2.28 (a) Make a vertical cut in a representative shell. (b) Open the shell up to form a flat plate.

In reality, the outer radius of the shell is greater than the inner radius, and hence the back edge of the plate would be slightly longer than the front edge of the plate. However, we can approximate the flattened shell by a flat plate of height f(xi*),f(xi*), width 2πxi*,2πxi*, and thickness ΔxΔx (Figure 2.28). The volume of the shell, then, is approximately the volume of the flat plate. Multiplying the height, width, and depth of the plate, we get

Vshellf(xi*)(2πxi*)Δx,Vshellf(xi*)(2πxi*)Δx,

which is the same formula we had before.

To calculate the volume of the entire solid, we then add the volumes of all the shells and obtain

Vi=1n(2πxi*f(xi*)Δx).Vi=1n(2πxi*f(xi*)Δx).

Here we have another Riemann sum, this time for the function 2πxf(x).2πxf(x). Taking the limit as nn gives us

V=limni=1n(2πxi*f(xi*)Δx)=ab(2πxf(x))dx.V=limni=1n(2πxi*f(xi*)Δx)=ab(2πxf(x))dx.

This leads to the following rule for the method of cylindrical shells.

Rule: The Method of Cylindrical Shells

Let f(x)f(x) be continuous and nonnegative. Define RR as the region bounded above by the graph of f(x),f(x), below by the x-axis,x-axis, on the left by the line x=a,x=a, and on the right by the line x=b.x=b. Then the volume of the solid of revolution formed by revolving RR around the y-axis is given by

V=ab(2πxf(x))dx.V=ab(2πxf(x))dx.
2.6

Now let’s consider an example.

Example 2.12

The Method of Cylindrical Shells 1

Define RR as the region bounded above by the graph of f(x)=1/xf(x)=1/x and below by the x-axisx-axis over the interval [1,3].[1,3]. Find the volume of the solid of revolution formed by revolving RR around the y-axis.y-axis.

Checkpoint 2.12

Define R as the region bounded above by the graph of f(x)=x2f(x)=x2 and below by the x-axis over the interval [1,2].[1,2]. Find the volume of the solid of revolution formed by revolving RR around the y-axis.y-axis.

Example 2.13

The Method of Cylindrical Shells 2

Define R as the region bounded above by the graph of f(x)=2xx2f(x)=2xx2 and below by the x-axisx-axis over the interval [0,2].[0,2]. Find the volume of the solid of revolution formed by revolving RR around the y-axis.y-axis.

Checkpoint 2.13

Define RR as the region bounded above by the graph of f(x)=3xx2f(x)=3xx2 and below by the x-axisx-axis over the interval [0,2].[0,2]. Find the volume of the solid of revolution formed by revolving RR around the y-axis.y-axis.

As with the disk method and the washer method, we can use the method of cylindrical shells with solids of revolution, revolved around the x-axis,x-axis, when we want to integrate with respect to y.y. The analogous rule for this type of solid is given here.

Rule: The Method of Cylindrical Shells for Solids of Revolution around the x-axis

Let g(y)g(y) be continuous and nonnegative. Define QQ as the region bounded on the right by the graph of g(y),g(y), on the left by the y-axis,y-axis, below by the line y=c,y=c, and above by the line y=d.y=d. Then, the volume of the solid of revolution formed by revolving QQ around the x-axisx-axis is given by

V=cd(2πyg(y))dy.V=cd(2πyg(y))dy.

Example 2.14

The Method of Cylindrical Shells for a Solid Revolved around the x-axis

Define QQ as the region bounded on the right by the graph of g(y)=2yg(y)=2y and on the left by the y-axisy-axis for y[0,4].y[0,4]. Find the volume of the solid of revolution formed by revolving QQ around the x-axis.

Checkpoint 2.14

Define QQ as the region bounded on the right by the graph of g(y)=3/yg(y)=3/y and on the left by the y-axisy-axis for y[1,3].y[1,3]. Find the volume of the solid of revolution formed by revolving QQ around the x-axis.x-axis.

For the next example, we look at a solid of revolution for which the graph of a function is revolved around a line other than one of the two coordinate axes. To set this up, we need to revisit the development of the method of cylindrical shells. Recall that we found the volume of one of the shells to be given by

Vshell=f(xi*)(πxi2πxi12)=πf(xi*)(xi2xi12)=πf(xi*)(xi+xi1)(xixi1)=2πf(xi*)(xi+xi12)(xixi1).Vshell=f(xi*)(πxi2πxi12)=πf(xi*)(xi2xi12)=πf(xi*)(xi+xi1)(xixi1)=2πf(xi*)(xi+xi12)(xixi1).

This was based on a shell with an outer radius of xixi and an inner radius of xi1.xi1. If, however, we rotate the region around a line other than the y-axis,y-axis, we have a different outer and inner radius. Suppose, for example, that we rotate the region around the line x=k,x=k, where kk is some positive constant. Then, the outer radius of the shell is xi+kxi+k and the inner radius of the shell is xi1+k.xi1+k. Substituting these terms into the expression for volume, we see that when a plane region is rotated around the line x=k,x=k, the volume of a shell is given by

Vshell=2πf(xi*)((xi+k)+(xi1+k)2)((xi+k)(xi1+k))=2πf(xi*)((xi+xi22)+k)Δx.Vshell=2πf(xi*)((xi+k)+(xi1+k)2)((xi+k)(xi1+k))=2πf(xi*)((xi+xi22)+k)Δx.

As before, we notice that xi+xi12xi+xi12 is the midpoint of the interval [xi1,xi][xi1,xi] and can be approximated by xi*.xi*. Then, the approximate volume of the shell is

Vshell2π(xi*+k)f(xi*)Δx.Vshell2π(xi*+k)f(xi*)Δx.

The remainder of the development proceeds as before, and we see that

V=ab(2π(x+k)f(x))dx.V=ab(2π(x+k)f(x))dx.

We could also rotate the region around other horizontal or vertical lines, such as a vertical line in the right half plane. In each case, the volume formula must be adjusted accordingly. Specifically, the x-termx-term in the integral must be replaced with an expression representing the radius of a shell. To see how this works, consider the following example.

Example 2.15

A Region of Revolution Revolved around a Line

Define RR as the region bounded above by the graph of f(x)=xf(x)=x and below by the x-axisx-axis over the interval [1,2].[1,2]. Find the volume of the solid of revolution formed by revolving RR around the line x=−1.x=−1.

Checkpoint 2.15

Define RR as the region bounded above by the graph of f(x)=x2f(x)=x2 and below by the x-axisx-axis over the interval [0,1].[0,1]. Find the volume of the solid of revolution formed by revolving RR around the line x=−2.x=−2.

For our final example in this section, let’s look at the volume of a solid of revolution for which the region of revolution is bounded by the graphs of two functions.

Example 2.16

A Region of Revolution Bounded by the Graphs of Two Functions

Define RR as the region bounded above by the graph of the function f(x)=xf(x)=x and below by the graph of the function g(x)=1/xg(x)=1/x over the interval [1,4].[1,4]. Find the volume of the solid of revolution generated by revolving RR around the y-axis.y-axis.

Checkpoint 2.16

Define RR as the region bounded above by the graph of f(x)=xf(x)=x and below by the graph of g(x)=x2g(x)=x2 over the interval [0,1].[0,1]. Find the volume of the solid of revolution formed by revolving RR around the y-axis.y-axis.

Which Method Should We Use?

We have studied several methods for finding the volume of a solid of revolution, but how do we know which method to use? It often comes down to a choice of which integral is easiest to evaluate. Figure 2.34 describes the different approaches for solids of revolution around the x-axis.x-axis. It’s up to you to develop the analogous table for solids of revolution around the y-axis.y-axis.

This figure is a table comparing the different methods for finding volumes of solids of revolution. The columns in the table are labeled “comparison”, “disk method”, “washer method”, and “shell method”. The rows are labeled “volume formula”, “solid”, “interval to partition”, “rectangles”, “typical region”, and “rectangle”. In the disk method column, the formula is given as the definite integral from a to b of pi times [f(x)]^2. The solid has no cavity in the center, the partition is [a,b], rectangles are vertical, and the typical region is a shaded region above the x-axis and below the curve of f(x). In the washer method column, the formula is given as the definite integral from a to b of pi times [f(x)]^2-[g(x)]^2. The solid has a cavity in the center, the partition is [a,b], rectangles are vertical, and the typical region is a shaded region above the curve of g(x) and below the curve of f(x). In the shell method column, the formula is given as the definite integral from c to d of 2pi times yg(y). The solid is with or without a cavity in the center, the partition is [c,d] rectangles are horizontal, and the typical region is a shaded region above the x-axis and below the curve of g(y).
Figure 2.34

Let’s take a look at a couple of additional problems and decide on the best approach to take for solving them.

Example 2.17

Selecting the Best Method

For each of the following problems, select the best method to find the volume of a solid of revolution generated by revolving the given region around the x-axis,x-axis, and set up the integral to find the volume (do not evaluate the integral).

  1. The region bounded by the graphs of y=x,y=x, y=2x,y=2x, and the x-axis.x-axis.
  2. The region bounded by the graphs of y=4xx2y=4xx2 and the x-axis.x-axis.
Checkpoint 2.17

Select the best method to find the volume of a solid of revolution generated by revolving the given region around the x-axis,x-axis, and set up the integral to find the volume (do not evaluate the integral): the region bounded by the graphs of y=2x2y=2x2 and y=x2.y=x2.

Section 2.3 Exercises

For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand.

114.

[T] Over the curve of y=3x,x=0,y=3x,x=0, and y=3y=3 rotated around the y-axis.y-axis.

115.

[T] Under the curve of y=3x,y=0,andx=3y=3x,y=0,andx=3 rotated around the y-axis.y-axis.

116.

[T] Over the curve of y=3x,y=0,andy=3y=3x,y=0,andy=3 rotated around the x-axis.x-axis.

117.

[T] Under the curve of y=3x,y=0,andx=3y=3x,y=0,andx=3 rotated around the x-axis.x-axis.

118.

[T] Under the curve of y=2x3,x=0,andx=2y=2x3,x=0,andx=2 rotated around the y-axis.y-axis.

119.

[T] Under the curve of y=2x3,x=0,andx=2y=2x3,x=0,andx=2 rotated around the x-axis.x-axis.

For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axisx-axis and are rotated around the y-axis.y-axis.

120.

y=1x2,x=0,andx=1y=1x2,x=0,andx=1

121.

y=5x3,x=0,andx=1y=5x3,x=0,andx=1

122.

y=1x,x=1,andx=100y=1x,x=1,andx=100

123.

y=1x2,x=0,andx=1y=1x2,x=0,andx=1

124.

y=11+x2,x=0,andx=3y=11+x2,x=0,andx=3

125.

y=sinx2,x=0,andx=πy=sinx2,x=0,andx=π

126.

y=11x2,x=0,andx=12y=11x2,x=0,andx=12

127.

y=x,x=0,andx=1y=x,x=0,andx=1

128.

y=(1+x2)3,x=0,andx=1y=(1+x2)3,x=0,andx=1

129.

y=5x32x4,x=0,andx=2y=5x32x4,x=0,andx=2

For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0y=0 around the x-axis.x-axis.

130.

y=1x2,x=0,x=1y=1x2,x=0,x=1 and the x-axis

131.

y=x2,x=0,x=2y=x2,x=0,x=2 and the x-axis

132.

y=x32,x=0,x=2,y=x32,x=0,x=2, and the x-axis

133.

y=2x2,x=1,x=2,y=2x2,x=1,x=2, and the x-axis

134.

x=11+y2,x=15,andy=0x=11+y2,x=15,andy=0

135.

x=1+y2y,y=1,y=4,x=1+y2y,y=1,y=4, and the y-axis

136.

x=cosy,y=0,andy=π x=cosy,y=0,andy=π

137.

x=y3-2y2,x=0,x=9,x=y3-2y2,x=0,x=9, and the y-axis

138.

x=y+1,x=1,x=3,x=y+1,x=1,x=3, and the x-axis

139.

x=27y3andx=3y4x=27y3andx=3y4

For the following exercises, find the volume generated when the region between the curves is rotated around the given axis.

140.

y=3x,y=0,x=0,andx=2y=3x,y=0,x=0,andx=2 rotated around the y-axis.y-axis.

141.

y=x3,x=0,andy=8y=x3,x=0,andy=8 rotated around the y-axis.y-axis.

142.

y=x2,y=x,y=x2,y=x, rotated around the y-axis.y-axis.

143.

y=x,y=0,andx=1y=x,y=0,andx=1 rotated around the line x=2.x=2.

144.

y=14x,x=1,andx=2y=14x,x=1,andx=2 rotated around the line x=4.x=4.

145.

y=xandy=x2y=xandy=x2 rotated around the y-axis.y-axis.

146.

y=xandy=x2y=xandy=x2 rotated around the line x=2.x=2.

147.

x=y3,x=1y,x=1,andx=2x=y3,x=1y,x=1,andx=2 rotated around the x-axis.x-axis.

148.

x=y2andy=xx=y2andy=x rotated around the line y=2.y=2.

149.

[T] Left of x=sin(πy),x=sin(πy), right of y=x,y=x, around the y-axis.y-axis.

For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume.

150.

[T] y=x2y=x2 and y=4xy=4x rotated around the y-axis.y-axis.

151.

[T] y=cos(πx),y=sin(πx),x=14,andx=54y=cos(πx),y=sin(πx),x=14,andx=54 rotated around the y-axis.y-axis.

152.

[T] y=x22x,x=2,andx=4y=x22x,x=2,andx=4 rotated around the y-axis.y-axis.

153.

[T] y=x22x,x=2,andx=4y=x22x,x=2,andx=4 rotated around the x-axis.x-axis.

154.

[T] y=3x32,y=x,andx=2y=3x32,y=x,andx=2 rotated around the x-axis.x-axis.

155.

[T] y=3x32,y=x,andx=2y=3x32,y=x,andx=2 rotated around the y-axis.y-axis.

156.

[T] x=sin(πy2)x=sin(πy2) and x=2yx=2y rotated around the x-axis.x-axis.

157.

[T] x=y2,x=y22y+1,andx=2x=y2,x=y22y+1,andx=2 rotated around the y-axis.y-axis.

For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures.

158.

Use the method of shells to find the volume of a sphere of radius r.r.

This figure has two images. The first is a circle with radius r. The second is a basketball.
159.

Use the method of shells to find the volume of a cone with radius rr and height h.h.

This figure has two images. The first is an upside-down cone with radius r and height h. The second is an ice cream cone.
160.

Use the method of shells to find the volume of an ellipse (x2/a2)+(y2/b2)=1(x2/a2)+(y2/b2)=1 rotated around the x-axis.x-axis.

This figure has two images. The first is an ellipse with a the horizontal distance from the center to the edge and b the vertical distance from the center to the top edge. The second is a watermelon.
161.

Use the method of shells to find the volume of a cylinder with radius rr and height h.h.

This figure has two images. The first is a cylinder with radius r and height h. The second is a cylindrical candle.
162.

Use the method of shells to find the volume of the donut created when the circle x2+y2=4x2+y2=4 is rotated around the line x=4.x=4.

This figure has two images. The first has two ellipses, one inside of the other. The radius of the path between them is 2 units. The second is a doughnut.
163.

Consider the region enclosed by the graphs of y=f(x),y=1+f(x),x=0,y=0,y=f(x),y=1+f(x),x=0,y=0, and x=a>0.x=a>0. What is the volume of the solid generated when this region is rotated around the y-axis?y-axis? Assume that the function is defined over the interval [0,a].[0,a].

164.

Consider the function y=f(x),y=f(x), which decreases from f(0)=bf(0)=b to f(1)=0.f(1)=0. Set up the integrals for determining the volume, using both the shell method and the disk method, of the solid generated when this region, with x=0x=0 and y=0,y=0, is rotated around the y-axis.y-axis. Prove that both methods approximate the same volume. Which method is easier to apply? (Hint: Since f(x)f(x) is one-to-one, there exists an inverse f−1(y).)f−1(y).)

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