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# Some of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding thevolumes 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) $V=\frac{1}{3} \pi h\left(2 R^{2}+r^{2}-\frac{2}{5} d^{2}\right)$

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Applications of Integration

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