   Chapter 5.2, Problem 71E

Chapter
Section
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

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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