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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 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
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Chapter 6 Solutions
Calculus: Early Transcendentals, Loose-leaf Version, 9th
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage