   Chapter 5.R, Problem 13E

Chapter
Section
Textbook Problem

# Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = cos 2 x ,    | x | ≤ π / 2 ,    y = 1 4 ; about x = π / 2

To determine

To set up:

An integral for the volume of a solid.

Explanation

1) Concept:

If x  is the radius of a typical shell, then circumference=2πx and height is y=f(x).

Then by shell method, the volume of the solid obtained by rotating the region under the curve y=f(x) about y- axis from a to b is

V= ab2πxf(x)dx;axb

where 2πx  is circumference, fx is height, and dx  is the thickness of the shell.

2) Given:

3) Calculation:

The region is bounded by

y=cos2x,  xπ2,  y= 14;  about x=π2

Draw the graph of the given curves.

The graph shows the region and representative strip to rotate about the line

x=π2

=π2-x

Using shell method, find the approximating shell with radius

π2-x

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