Chapter 6.2, Problem 85E

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 ‒ cx², ‒ 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 = ch²/ 4.

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

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