Explanation Given: The curves are y = x 2 and y = 8 x , then these curves is rewrite in terms of y are x = ( y ) 1 2 , x = y 2 8 . The region is revolved about the both axis. Formula used: The volume of the solid (about the y axis) is ∫ a b π ( f ( y ) 2 − g ( y ) 2 ) d y , where f ( y ) and g ( y ) are mentioned below. The volume of the solid (about the x axis) is ∫ c d π ( f ( x ) 2 − g ( x ) 2 ) d x , where f ( x ) and g ( x ) are mentioned below. Calculation: Calculate the volume of the solid when the region R is revolving about the- x axis . The curves are y = x 2 and y = 8 x . Let f ( x ) = 8 x and g ( x ) = x 2 . Let l = f ( x ) 2 − g ( x ) 2 . Obtain the volume of the solid, first find the value of l with f ( x ) = 8 x and g ( x ) = x 2 as follows, l = ( 8 x ) 2 − ( x 2 ) 2 = 8 x − x 4 Obtain the limit of integration, it is suffices to find the intersection point of the curves f ( x ) = 8 x and g ( x ) = x 2 . 8 x = x 2 8 x = x 4 x 4 − 8 x = 0 x = 0 , x = 2 Substitute the value of l = 8 x − x 4 in the above mentioned volume of the solid formula (about the x axis) and take x = 0 as lower limit and x = 2 as upper limit. V = ∫ 0 2 π ( 8 x − x 4 ) d x = 8 π ∫ 0 2 x d x − π ∫ 0 2 x 4 d x = 8 π 2 [ x 2 ] 0 2 − π 5 [ x 5 ] 0 2 = 4 π ( 2 2 − 0 ) − π 5 ( 2 5 − 0 ) On further simplification, the above equation becomes, V = 4 π ( 4 − 0 ) − π 5 ( 32 − 0 ) = 16 π − 32 π 5 = 80 π − 32 π 5 = 48 π 5 Hence, the volume of the solid about the x axis is 48 π 5

Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book

2nd Edition

ISBN: 9780321954237

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 6.3, Problem 44E

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The greater volume of the solid when the region R is revolving about the coordinate axes (the x axis and the y axis).

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Chapter 6.3, Problem 43E

Chapter 6.3, Problem 45E

Chapter 6 Solutions

Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book

Ch. 6.1 - Prob. 1QC Ch. 6.1 - Prob. 2QC Ch. 6.1 - Prob. 3QC Ch. 6.1 - Prob. 4QC Ch. 6.1 - Prob. 5QC Ch. 6.1 - Prob. 6QC Ch. 6.1 - Explain the meaning of position, displacement, and...Ch. 6.1 - Suppose the velocity of an object moving along a...Ch. 6.1 - Given the velocity function v of an object moving...Ch. 6.1 - Explain how to use definite integrals to find the...

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The greater volume of the solid when the region R is revolving about the coordinate axes (the x axis and the y axis).

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The greater volume of the solid when the region R is revolving about the coordinate axes (the x axis and the y axis).

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