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 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 = 1/3 πh(2R2 + r2 - 2/5 d2)