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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

x = 0 ,   x = 9 y 2 ; about the 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 and sketch the region, the solid, and a typical disk or washer.

Explanation

1) Concept:

i. If the cross section is a washer with the inner radius rin and outer radius rout, then the area of the 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 solid revolution about y-axis

V= abA(y)dy

2) Given:

The region is bounded by x=0,x=9-y2 rotated about the line x=-1

3) Calculation:

As the region is bounded by x=0,x=9-y2 rotated about the line x=-1,

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

A cross section of the solid is the washer with outer radius is 9-y2--1= 10-y2 (distance from x=-1 to x=9-y2)  and inner radius is 0--1=1, which is the distance from x=-1 to x=0.

So its area is given by

Ay=πouter radius2-πinner radius2

=π10-y22-π12

=π100-20y2+y4-1

A(x)=π99-20y2+y4

The region of integration is bounded by x=0,x=9-y2

Therefore, 9-

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