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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

Use cylindrical shells to find the volume of the solid.

A sphere of radius r

To determine

To find:

The volume of the sphere of the radius r.

Explanation

1) Concept:

i. If x is the radius of the typical shell, then the circumference =2πx and the height is y.

ii. By the shell method, the volume of the solid by rotating the region under the curve y=f(x) about y- axis from a to b is

V= ab2πx f(x)dx

where,  0ab

2) Given:

The radius of the sphere is r.

3) Calculation:

A sphere is obtained by rotating the region under the curve x2+y2=r2 of the first and the fourth quadrant about y-axis.

Hus we have y2=r2-x2

 y= ±r2-x2

Let the given region rotate about the y- axis.

Using the shell method, find the typical approximating shell with the radius x.

The figure gives top half of height of a typical shell. By symmetry therefore, the circumference is 2πx and the height is y= r2-x2--r2-x2=2r2-x2

So, the total volume is

V= ab2πx 2r2-x2dx

V=- 2π0r(-2x)r2-x<

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