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 - cx^2 $, $ \frac{-h}{2} \le x \le \frac{h}{2} $, where c is a positive constant. Show that the radius of each end of the barrel is $ r = R - d $, where $ d = \frac{ch^2}{4} $.

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

$$ V = \frac{1}{3} \pi h (2R^2 + r^2 - \frac{2}{5} d^2) $$

a) $R-d=r$

b) $V=\frac{1}{3} \pi h\left(2 R^{2}+r^{2}-\frac{2}{5} d^{2}\right)$

Applications of Integration

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Harvey Mudd College

University of Nottingham

Idaho State University

Boston College

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