   Chapter 5.2, Problem 38E

Chapter
Section
Textbook Problem

# Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x 2 − 2 x ,   y = x cos ( π x / 4 ) ;    about y = 2

To determine

To find:

The exact volume of the solid obtained by rotating the region bounded by the given curves about the line y= 2 by using a computer algebra system.

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 of revolution about x-axis is

V= abA(x)dx

2) Given:

3) Calculation:

Bounded region of the given curves is shown below:

From the above graph

x coordinate of the points of intersection: x=0 and  x= 2

The cross section is perpendicular to the x-axis, and it is a washer.

Thus, the outer radius is the distance from the curve y=x2-2x to the axis of rotation y= 2 and the inner radius is the distance from the curve y=xcosπx4 to the axis of rotation  y= 2

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