# 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 1715 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

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 1715 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 -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.

Show that the volume enclosed by the barrel is

(a) A barrel with height and maximum radius R is constructed by rotating about the -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.

Show that the volume enclosed by the barrel is

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