   Chapter 5.3, Problem 43E

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 = ( y − 1 ) 2 ,   x − y = 1 ;    about  x = − 1

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 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,

ii. The volume of solid revolution about y-axis,

V= abA(y)dy

2) Given:

The region bounded by x=y-12, x-y=1  rotated about the x=-1

3) Calculation:

Find the volume by using washers.

The graph of the region bounded by the given curves is

From figure, as region rotate about the line x=-1, so strip is perpendicular to y-axis.

A cross section of the solid is washer with outer radius is 1+y--1= y+2  and inner radius is y-12-(-1)

From the graph, the points of intersection are (1, 0) and (4, 3)

So its area is given by,

Ay=2+y2-(y-12-(-1)) 2

V= 03Aydy=03[π2+y

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