Solved: Some of the pioneers of calculus, such as Kepler and Newton, were inspired by

Chapter 6, Problem 6.2.71

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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 is constructed by rotating about the -axis the parabola y R cx , , where is a positive h constant. Show that the radius of each end of the barrel is, where .(b) Show that the volume enclosed by the barrel is V 13h(2R2 r 2 25 d2 )d

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