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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

Some of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding the volumes of wine barrels. (In fact Kepler published a book Stereometria doliorum in 1615 devoted to methods for finding the volumes of barrels.) They often approximated the shape of the sides by parabolas.

(a) A barrel with height h and maximum radius R is constructed by rotating about the x-axis the parabola y = R c x 2 , h / 2 x h / 2 , where c is a positive constant. Show that the radius of each end of the barrel is r = R d , where d = c h 2 / 4.

(b) Show that the volume enclosed by the barrel is

v = 1 3 π h ( 2 R 2 + r 2 2 5 d 2 )

To determine

(a)

To show:

Radius of each end of the barrel is r=R-d

Explanation

i) Solid of Revolution:

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

V=abA(x)dx or V=cdA(y)dy

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

In case of thewasher method, find the inner and outer radius and compute the area of thewasher 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=20

To determine

(b)

To show:

Volume enclosed by thebarrel is V=13πh2R2+r2-25d2

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