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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

A hole of radius r is bored through the middle of a cylinder of radius R > r at right angles to the cylinder. Set up, but do not evaluate, an integral for the volume cut out.

To determine

To find:

Integral for the volume of cut out

Explanation

1) Concept:

i) Solid of Revolution:

Volume of solid revolution revolving a region around a line is given by

V=abA(x)dx or V=cdA(y)dy

In case of the disk method, the radius is found in terms of x or y and use A=π·radius2

In case of the washer method, find the inner and outer radius and compute the area of washer by subtracting the area of inner disk from outer disk, use

A=π·outer radius2-π·inner radius2

ii) Integrals of Symmetric functions:

Suppose f is continuous on [-a, a]

a) If f is even [f-x=fx], then -aaf(x)dx=20af(x)dx

b) If f is odd [f-x=-fx], then -aaf(x)dx=0

2) Calculation:

Now from the front side, the circle appears to be in a y-z plane,

Therefore, equation of circle is y2+z2=R2

From the top side, the hole lies on a x-y plane,

Therefore, equation of circle is x2+y2=r2

From the front side, the diagram will appear as shown below:

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