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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

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 + 4 y 2 = 4

(a) About y = 2

(b) About x = 2

To determine

Part (a):

i. To set up:

An integral for the volume of the solid obtained by rotating the region bounded by the given curves about the line  y=2.

ii. To evaluate:

The integral by using the calculator up to five decimal places

Explanation

1) Concept:

i. If the cross section is a washer with inner radius rin and outer radius rout, then area of washer is obtained by subtracting the area of the inner disk from the area of the outer disk,

A=π outer radius2 -π inner radius2

ii. The volume of a solid revolution about the  x-axis is

V= abA(x)dx

2) Given:

x2+4y2=4, about  y=2

3) Calculation:

Bounded region of the given curves is shown below:

The region rotates about the line =2, so the cross-section is perpendicular to the x-axis, and it forms a washer.

To find the outer and inner radius, solve the given equation x2+4y2=4 for y since the limits of integration are in terms of  x

x2+4y2=4

Divide both sides by  4

x24+y2=1

Subtract x24 from both sides

y2=1-x24

By taking square root of both sides

y=±1-x24

So the outer radius is the distance of the bottom half of the ellipse to the axis of rotation y=2

Outer radius=2--1-x24=2+1-x24

The inner radius is the distance of the top half of the ellipse to the axis of rotation y=2

Inner radius =2-1-x24

The solid lies between x=-2  and  x=2

To determine

Part (b):

i. To set up:

An integral for the volume of the solid obtained by rotating the region bounded by the given curves  about the line  x=2.

ii. To evaluate:

The integral by using the calculator up to five decimal places

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