(a) Explanation Formula used: The volume of the ellipse if an ellipse is rotated about its major axis is V = ∫ − a a π y 2 d x . Calculation The ellipse is rotated about its major axis. The equation of the ellipse is x 2 a 2 + y 2 b 2 = 1 . Rewrite the above equation as, y 2 b 2 = 1 − x 2 a 2 y 2 = ( 1 − x 2 a 2 ) × b 2 y = ( 1 − x 2 a 2 ) × b 2 y = ( a 2 − x 2 a 2 ) × b 2 On further simplifications, y = b 2 a 2 ( a 2 − x 2 ) y = ± b a a 2 − x 2 Obtain the volume of the ellipse if it rotated about it major axis as follows. Substitute y = ± b a a 2 − x 2 in V = ∫ − a a π y 2 d x b) To determine The volume of the resulting solid if an ellipse is rotated about its minor axis.

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Chapter 10.5, Problem 64E

(a)

To determine

The volume of the resulting solid if an ellipse is rotated about its major axis.

To determine

The volume of the resulting solid if an ellipse is rotated about its minor axis.

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Chapter 10.5, Problem 63E

Chapter 10.5, Problem 65E

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The volume of the resulting solid if an ellipse is rotated about its major axis.

Solution Summary: The author calculates the volume of the resulting solid if an ellipse is rotated about its major axis.

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The volume of the resulting solid if an ellipse is rotated about its major axis.

The volume of the resulting solid if an ellipse is rotated about its minor axis.

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