Chapter 5.2, Problem 71E

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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
*c* is a positive constant. Show that the radius of each end of the barrel is

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

To determine

**(a)**

**To show:**

Radius of each end of the barrel is

Explanation

i) Solid of Revolution:

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

In case of thedisk method, radius is found in terms of x or y, and use

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

ii) Integrals of Symmetric functions:

Suppose

a) If

To determine

**(b)**

**To show:**

Volume enclosed by thebarrel is

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