Explanation Given information: The region formed is bounded by the curves y = x tan − 1 x and the line y = 0 for within the region given by the limits 0 ≤ x ≤ 1 . Formula: ∫ a b x n d x = [ x n + 1 n + 1 ] a b ∫ u ( x ) v ′ ( x ) d x = u ( x ) v ( x ) − ∫ v ( x ) u ′ ( x ) d x (1) d d x ( tan − 1 x ) = 1 1 + x 2 ∫ a b 1 1 + x 2 d x = [ tan − 1 x ] a b Calculation: Calculate the volume of the solid formed by revolving the region about the x axis, V using the below expression. V = ∫ 0 1 π y 2 d x Substitute x tan − 1 x for y . V = ∫ 0 1 π ( x tan − 1 x ) 2 d x V = π ∫ 0 1 x tan − 1 x d x (2) Apply trigonometric substitution by assuming the variable u ( x ) as tan − 1 x . u ( x ) = tan − 1 x Differentiate the equation with respect to x . u ′ ( x ) = 1 1 + x 2 Apply integral substitution by assuming the term x as v ′ ( x ) . v ′ ( x ) = x Integrate the above equation with respect to x . ∫ d v = ∫ x d x v ( x ) = x 2 2 Substitute tan − 1 x for u ( x ) , x 2 2 for v ( x ) , 1 1 + x 2 for u ′ ( x ) , and x for v ′ ( x ) with the limits in equation (1) as below

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ISBN: 9780134175706

Author: Unknown

Publisher: PEARSON

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Chapter 8.3, Problem 56E

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The volume of the solid formed by revolving this region about the x axis.

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Chapter 8.3, Problem 55E

Chapter 8.3, Problem 57E

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