   Chapter 5.2, Problem 42E

Chapter
Section
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,

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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