   Chapter 5.2, Problem 7E

Chapter
Section
Textbook Problem

# Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = x 3 ,   y = x ,   x ≥ 0 ; about the x-axis

To determine

To find:

The volume of the solid obtained by rotating the region bounded by the given curves about the x axis and sketch the region, the solid, and a typical disk or washer.

Explanation

1) Concept:

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

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

V= abA(x)dx

2) Given:

y=x3, y=x ,  x0; about the x- axis

3) Calculation:

The region is bounded by y=x3, y=x,  x0 and rotated about the x- axis is shown below.

Here, the region is rotated about x– axis, so the cross-section is perpendicular to x-axis.

A cross section of the solid is thewasher with outer radius x and inner radius x3.

So, its cross sectional area becomes

=π(x2-x6)

Here, the region of integration is bounded by y=x3, y=x,  x0.

At the point of intersection of both curves

x3=x

x3-x=0

xx2-1=0

x=0, x=1  since x0

Therefore, the solid lies between  x=0 and x= 1

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