Explanation Given: The solid region is formed by the equation of the curves is f ( x ) = 2 x − x 2 , g ( x ) = x . The region R is bounded by the above curves. Formula used: (1) The volume of the solid (shell method) is V = ∫ a b 2 π x ( f ( x ) − g ( x ) ) d x , where 2 π x and f ( x ) − g ( x ) are the shell circumference and height, respectively. (2) The volume of the solid (washer method) is V = ∫ c d π ( p ( y ) 2 − q ( y ) 2 ) d y , where p ( y ) and q ( y ) are the outer and inner radius, respectively. Calculation: Calculate the volume of the solid using shell method as follows, Let l = f ( x ) − g ( x ) . Calculate the value of l with g ( x ) = x and f ( x ) = 2 x − x 2 . l = ( 2 x − x 2 ) − x = 2 x − x 2 − x = x − x 2 To find the limit of the integration, it is enough to find the intersection points of the curve f ( x ) = 2 x − x 2 and g ( x ) = x as follows, 2 x − x 2 = x x − x 2 = 0 x ( 1 − x ) = 0 x = 0 , x = 1 Substitute the value of f ( x ) − g ( x ) = ( x − x 2 ) in the above mentioned formula and take x = 0 as lower limit and x = 1 as upper limit to calculate the volume of the solid. V = ∫ 0 1 2 π x ( x − x 2 ) d x = 2 π ∫ 0 1 ( x 2 − x 3 ) d x = 2 π ∫ 0 1 x 2 d x − 2 π ∫ 0 1 x 3 d x = 2 π [ x 3 3 ] 0 1 − 2 π [ x 4 4 ] 0 1 On further simplification, V = 2 π 3 ( 1 − 0 ) − π 2 ( 1 − 0 ) = 2 π 3 − π 2 = 4 π − 3 π 6 = π 6 Thus, the volume of the solid using shell method is π 6

Single Variable Calculus Format: Unbound (saleable)

3rd Edition

ISBN: 9780134765761

Author: Briggs, William L.^cochran, Lyle^gillett, Bernard^

Publisher: Prentice Hall

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Chapter 6.4, Problem 3QC

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To comment: The method (washer or shell) which makes to an integral easier when R is revolved about the y axis.

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Chapter 6.4, Problem 2QC

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To comment: The method (washer or shell) which makes to an integral easier when R is revolved about the y axis.

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