   Chapter 5.3, Problem 40E

Chapter
Section
Textbook Problem

# The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. y 2 − x 2 = 1 ,   y = 2 ;    about the  y -axis

To determine

To find:

The volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

Explanation

1) Concept:

The volume of the disk obtained by the revolution about the y-axis is

V= abA(y)dy where A(y) is the area of cross-section perpendicular y-axis.

2) Given:

The region bounded by y2-x2=1, y=2,  rotated about the y- axis.

3) Calculation:

The graph of the region bounded by the given curves is

Find the volume by using the area of disks which are perpendicular to y-axis,

Here, the region is bounded by  y2-x2=1x=±y2-1, y=2

We are interested only in positive y and the region is bounded in y=[1, 2]. So

V=12πy2-12dx

=12π

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