(a) Explanation Given: The region is bounded by f ( x ) = x − p and the x axis on the interval [ 1 , ∞ ) . The region is revolved about the x axis and equation of x axis is y = 0 . Let y = g ( x ) , then g ( x ) = 0 . Formula used: The volume of the solid (disk method) is ∫ a b π ( f 2 ( x ) − g 2 ( x ) ) d x , where f ( x ) and g ( x ) is mentioned above. Definition used: Improper integral “If f is continuous on [ a , ∞ ) , then ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x .” Theorem used: Fundamental theorem of calculus “If f is continuous on [ a , b ] and F is anti-derivative of f , then ∫ a b f ( x ) d x = F ( b ) − F ( a ) .” Calculation: The region is bounded by f ( x ) = x − p and g ( x ) = 0 . The function f ( x ) = x − p is unbounded at the point x = 0 on the interval [ 1 , ∞ ) . Let l = f 2 ( x ) − g 2 ( x ) . To find the volume of the solid, first find the value of l with f ( x ) = x − p and g ( x ) = 0 as follows, l = ( x − p ) 2 − 0 2 = x − 2 p Substitute the value of l = x − 2 p in the above mentioned volume of the solid formula and take a = 1 as lower limit and b = ∞ as upper limit. V = π ∫ 1 ∞ x − 2 p d x To find the value of p such that volume of the solid exist following three cases arise. That is, Case 1: when 2 p > 1 Since 2 p > 1 , this implies 2 p − 1 > 0 and use it in the above calculated volume integral. That is, Evaluate the volume integral V = π ∫ 1 ∞ x − 2 p d x through the definition and theorem mentioned above. V = π ∫ 1 ∞ x − 2 p d x = π lim b → ∞ ∫ 1 b x − 2 p d x ( Definition used ) = − π lim b → ∞ [ x − ( 2 p − 1 ) 2 p − 1 ] 1 b = − π 2 p − 1 lim b → ∞ [ ( b ) x − ( 2 p − 1 ) − ( 1 ) x − ( 2 p − 1 ) ] ( Theorem used ) On further simplification, the volume becomes, V = − π (b) To determine To find: The value of p such that the volume of the solid exist when the region R is revolved about the y axis.

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ISBN: 9781323178522

Author: Briggs

Publisher: PEARSON

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Chapter 7.8, Problem 76E

(a)

To determine

To find: The value of p such that the volume of the solid exists when the region R is revolved about the x axis.

(b)

To determine

To find: The value of p such that the volume of the solid exist when the region R is revolved about the y axis.

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Chapter 7.8, Problem 75E

Chapter 7.8, Problem 77E

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The value of p such that the volume of the solid exists when the region R is revolved about the x axis.

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