Explanation Given: The given circle ( x − h ) 2 + y 2 = r 2 , h > r . Formula used: The trigonometric identity is: cos 2 θ = 1 − sin 2 θ If f ( x ) is an odd function. Then, ∫ − a a f ( x ) d x = 0 Calculation: Consider the followingcircle ( x − h ) 2 + y 2 = r 2 . From the above equation, this circle has centred at ( h , 0 ) and has radius r . Now use cylindrical method, to get x = h − r and x = h + r as the boundary of integration. The cylinder’s width is 2 y = 2 r 2 − ( x − h ) 2 and length is 2 π x and the thickness is d x . Now, the volume V , which is given as: V = ∫ h − r h + r 4 π x r 2 − ( x − h ) 2 d x V = 4 π ∫ h − r h + r x r 2 − ( x − h ) 2 d x and, d V = 4 π x r 2 − ( x − h ) 2 Hence, the total volume is V = ∫ 2 4 4 π x r 2 − ( x − h ) 2 d x . As r 2 − ( x − h ) 2 is of the form a 2 − x 2 ; Let x − h = r sin θ ; Differentiating it, gives; d x = r cos θ d θ Now to change the limit of integration; Substitute x = h − r to get, sin θ = − 1 θ = − π 2 Substitute x = h + r to get, sin θ = 1 θ =

10th Edition

ISBN: 9781285057095

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 8.4, Problem 60E

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To calculate: The volume of the torus generated by revolving the region bounded by the graph of the given circle about the y axis.

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Chapter 8.4, Problem 59E

Chapter 8.4, Problem 65E

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The volume of the torus generated by revolving the region bounded by the graph of the given circle about the y axis.

Solution Summary: The author calculates the volume of the torus generated by revolving the region bounded by the graph of a given circle about the y axis.

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To calculate: The volume of the torus generated by revolving the region bounded by the graph of the given circle about the y axis.

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