Explanation Determine the radius of the disk obtained by rotation to determine the volume of the solid. Given: The solid generated by rotating the region inside the ellipse with equation ( x a ) 2 + ( y b ) 2 = 1 around the x -axis is called an ellipsoid. Calculation: Rotating the ellipse about the x -axis produces an ellipsoid whose cross-sections are disks whose radius is given by R = b 1 − ( x / a ) 2 . Therefore, the volume of the ellipsoid will be π ∫ − a a ( b 1 − ( x a ) 2 ) 2 d x = π b 2 ∫ − a a ( 1 − x 2 a 2 ) d x = π b 2 [ x − x 3 3 a 2 ] − a a = 4 3 π a b 2 Rotating the ellipse about the y -axis produces an ellipsoid whose cross-sections are disks...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 6.3, Problem 61E

To determine

To show that the ellipsoid has volume $\frac{4}{3} π a b^{2}$ and also determine the volume, if the ellipse is rotated around the $y$ -axis

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Chapter 6.3, Problem 60E

Chapter 6.3, Problem 62E

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Chapter 6 Solutions

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To show that the ellipsoid has volume 4 3 π a b 2 and also determine the volume, if the ellipse is rotated around the y -axis

Solution Summary: The author proves that the ellipsoid has volume 4 3 a b 2 and determines the volume of the solid.

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To show that the ellipsoid has volume $\frac{4}{3} π a b^{2}$ and also determine the volume, if the ellipse is rotated around the $y$ -axis

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Chapter 6 Solutions

To show that the ellipsoid has volume 4 3 π a b 2 and also determine the volume, if the ellipse is rotated around the y -axis

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To show that the ellipsoid has volume $\frac{4}{3} π a b^{2}$ and also determine the volume, if the ellipse is rotated around the $y$ -axis