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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 line. Sketch the region, the solid, and a typical disk or washer.

y = 1 + sec x ,   y = 3 ; about y = 1

To determine

To find:

The volume of the solid obtained by rotating the region bounded by the given curves about the  line y=1 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 the 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 x-axis is

V= abA(x)dx

2) Given:

The region is bounded by y=1+secx, y=3; about y=1.

3) Calculation:

The region is bounded by y=1+secx, y=3 rotated about the line y=1 is shown below.

Here, the region is rotated about the line y=1, so the cross-section is perpendicular to the x-axis.

A cross section of the solid is the washer with the outer radius 3-1=2 and the inner radius is 1+secx-1=secx

So, its cross sectional area is

Ax=πouter radius2-πinner radius2 =π22-πsecx2

=π4-sec2x

The region of integration is bounded by y=1+secxand y=3

At intersection of these curves

1+secx=3 

secx=2

Hence,

x= -π3  or  x=π3

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