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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

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

π 1 4 [ 3 2 ( 3 x ) 2 ] d x

To determine

To describe:

The solid for an integral π1432-3-x2dx which represents the volume of a solid

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 the solid revolution about the x-axis is

V= abA(x)dx

2) Given:

π1432-3-x2dx

3) Calculation:

Compare the given integral with expression for volume.

Thus we see that the cross section of the solid of volume the integral represents, is a washer with outer radius 3  and inner radius 3- x, and it is perpendicular to x- axis. So the axis of rotation is x-axis

So, the resulting solid is obtained by rotating the area bounded by the line y=3 and the curve y=3-x and it lies between  x=1  and  x=4, about the x-axis

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