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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Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.5, Problem 61E

To determine

Formula for the volume V of the given torus.

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Chapter 16.5, Problem 60E

Chapter 16.5, Problem 62E

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Chapter 16 Solutions

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Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

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Prob. 35CRE Ch. 16 - Prob. 36CRE Ch. 16 - Prob. 37CRE Ch. 16 - Prob. 38CRE Ch. 16 - Prob. 39CRE Ch. 16 - Prob. 40CRE Ch. 16 - Prob. 41CRE Ch. 16 - Prob. 42CRE Ch. 16 - Prob. 43CRE Ch. 16 - Prob. 44CRE Ch. 16 - Prob. 45CRE Ch. 16 - Prob. 46CRE Ch. 16 - Prob. 47CRE Ch. 16 - Prob. 48CRE Ch. 16 - Prob. 49CRE Ch. 16 - Prob. 50CRE Ch. 16 - Prob. 51CRE Ch. 16 - Prob. 52CRE Ch. 16 - Prob. 53CRE Ch. 16 - Prob. 54CRE Ch. 16 - Prob. 55CRE Ch. 16 - Prob. 56CRE Ch. 16 - Prob. 57CRE Ch. 16 - Prob. 58CRE Ch. 16 - Prob. 59CRE Ch. 16 - Prob. 60CRE Ch. 16 - Prob. 61CRE

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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