Explanation Given information: The disk x 2 + y 2 ≤ a 2 is revolved about the line x = b ( b > a ) . The value of ∫ − a a a 2 − y 2 d y = π a 2 2 . Calculation: Provide the equation of line as shown below. x 2 + y 2 ≤ a 2 x 2 = a 2 − y 2 x = a 2 − y 2 Sketch the disk x 2 + y 2 ≤ a 2 is revolved about the line x = b ( b > a ) as shown in Figure 1. Refer to Figure 1. The outer radius R ( y ) = b + a 2 − y 2 . The inner radius r ( y ) = b − a 2 − y 2 . The region lies between − a to a . Calculate the volume of the solid using the formula: V = ∫ a b π ( [ R ( y ) ] 2 − [ r ( y ) ] 2 ) d y (1) Substitute b + a 2 − y 2 for R ( y ) , b − a 2 − y 2 for r ( y ) , − a for a , and a for b in Equation (1)

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 6.1, Problem 57E

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Calculate the volume of a torus.

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University Calculus: Early Transcendentals (4th Edition)

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