   Chapter 5.3, Problem 41E

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. x 2 + ( y − 1 ) 2 = 1 ;    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:

i. If the cross section is the disc and the radius of the disc is in terms of x or y, then

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

V= abA(y)dy

2) Given:

The region bounded by x2+y-12=1,  rotated about the y- axis.

3) Calculation:

Find the volume by using the disks.

The graph of the region bounded by the given curves is

x2+y-12=1

Solve for x from this

x=±1-y-12

A cross section of the solid is the disk with the radius 1-y-12

So, its area is given by

Ay=π1-y-122

V= 02Aydy=02π1-

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