Explanation The torus is obtained as a volume of revolution, and this volume is computed by applying Pappus’ Theorem . Given: The torus as a solid of revolution by revolving the circle given below about the x-axis. Formula used: Volume V is given by (Pappus) V = 2 π ∫ x = c x = d ( ∫ g ( x ) h ( x ) y d y ) d x (1) Here V is the volume of revolution obtained by revolving the region between the appropriate limits g(x) and h(x) ( ( c ≤ x ≤ d ) Calculation: (1) From the diagram, c = − b , d = b (2) The equation the circle is x 2 + ( y − a ) 2 = b 2 ⇒ y = a ± b 2 − x 2 (3) So, g ( x ) = a − b 2 − x 2 , h ( x ) = a + b 2 − x 2 (4) Using (1), (2) and (4), the required volume is V = 2 π ∫ x = − b x = b ( ∫ a − b 2 − x 2 a + b 2 − x 2 y

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ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.5, Problem 63E

To determine

Formula for the volume V of the given torus.

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Chapter 16.5, Problem 62E

Chapter 16.5, Problem 64E

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Chapter 16 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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Formula for the volume V of the given torus.

Solution Summary: The torus is obtained as a volume of revolution by revolving the circle given below about the x-axis.

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Formula for the volume V of the given torus.

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Chapter 16 Solutions