To determine The volume of the solid generated when R is revolved about the y axis. Explanation Given: The region bound by the curves are y = 1 x 2 ( x 2 + 2 ) 2 and y = 0 with x = 1 and x = 2 . The region R is revolving about the y axis. Formula used: The volume of the solid (shell method) is V = ∫ a b 2 π x ( f ( x ) − g ( x ) ) d x . Simple linear factor : A factor ( x − r ) in the denominator requires the partial fractions A ( x − r ) . Repeated linear factor : A factor ( x − r ) m with m > 1 in the denominator requires the partial fractions A 1 ( x − r ) + A 2 ( x − r ) 2 + ... + A m ( x − r ) m . Calculation: The region R is bounded by two curves. Let f ( x ) = 1 x 2 ( x 2 + 2 ) 2 and g ( x ) = 0 with f ( x ) ≥ g ( x ) and l = f ( x ) − g ( x ) . Obtain the value of l . l = 1 x 2 ( x 2 + 2 ) 2 − 0 = 1 x 2 ( x 2 + 2 ) 2 Obtain the volume over the interval [ 1 , 2 ] . Let u = x 2 , then d u = 2 x d x 1 2 d u = x d x V = 2 π [ ∫ 1 2 1 2 ( 1 u ( u + 2 ) 2 ) d u ] . (1) Note that, in the denominator u is in the form of simple linear factor and ( u + 2 ) 2 is in the form of repeated linear factor. So, use the above result to obtain the required result. Simplify the expression 1 u ( u + 2 ) 2 using partial fraction decomposition. 1 u ( u + 2 ) 2 = A u + B ( u + 2 ) + C ( u + 2 ) 2 (2) 1 u ( u + 2 ) 2 = A ( u + 2 ) 2 + B ( u ( u + 2 ) ) + C u u ( u + 2 ) 2 That is, 1 = A ( u + 2 ) 2 + B ( u ( u + 2 ) ) + C u (3) Simplify the equation (2). 1 = A ( u + 2 ) 2 + B ( u ( u + 2 ) ) + C u = A u 2 + 4 A u + 4 A + B u 2 + 2 B u + C u = u 2 ( A + B ) + u ( 4 A + 2 B + C ) + 4 A Equate the co-efficient of u 2 in equation (3), A + B = 0 That is, A + B = 0 . (4) Equate the co-efficient of u in equation (3), That is, 4 A + 2 B + C = 0 (5) Equate the co-efficient of constant in equation (3), 4 A = 1 A = 1 4 Thus, the value of A = 1 4 . Substitute A = 1 4 in equation (4), 1 4 + B = 0 B = − 1 4 Thus, the value of B = − 1 4

MyLab Math with Pearson eText -- Standalone Access Card -- for Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134856926

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

Publisher: PEARSON

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Chapter 8.5, Problem 70E

To determine

The volume of the solid generated when R is revolved about the y axis.

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MyLab Math with Pearson eText -- Standalone Access Card -- for Calculus: Early Transcendentals (3rd Edition)

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