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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 / x ,   y = 0 ,   x = 1 ,   x = 4 ; about the x-axis

To determine

To find:

The volume of the solid obtained by rotating the region bounded by the given curves about the x  axis and sketch the region, the solid and a typical disk or washer.

Explanation

1) Concept:

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

A=π radius2

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

V= abA(x)dx

2) Given:

y=1x, y=0, x=1, x=4; About the x- axis

3) Calculation:

The solid obtained by rotating the region bounded by

y=1x, y=0, x=1, x=4 about the x- axis is as shown below:

Here the region is rotated about the x – axis, so the cross-section is perpendicular to the x-axis.

A cross section of a solid is disk with radius 1x ,

So its cross sectional area becomes,

Ax=π1x2=πx-2

The solid lies between  x=1

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