   Chapter 5.2, Problem 15E

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 = 0 ,   x = 1 ; about x = 2

To determine

To find:

The volume of solid obtained by rotating the region bounded by the given curves about the line x=2 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 revolution about the y-axis is

V= abA(y)dy

2) Given:

y=x3, y=0,x=1; about the line x=2

3) Calculation:

The region  bounded by y=x3, y=0,x=1 and the solid obtained by rotation about the line x=2 are shown below.

Here, the region is rotated about the line x=2, so the cross-section is perpendicular to y-axis.

A cross section of the solid is the washer with the outer radius 2-y3 and the inner radius is 2-1=1.

So, its cross sectional area is

=π4-4y13+y23-1

A(y)=π3-4y13+y23

The region of integration is bounded by y=x3, y=0 and x=1

So one boundary point is given by,

y=13

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