Explanation Given: y = f ( x ) is a decreasing function on [0, b ] such that f ( b ) = 0 . Proof: Since y = f ( x ) is a decreasing function on [0, b ] such that f ( b ) = 0 . ⇒ d y d x = d d x ( f ( x ) ) = f ′ ( x ) < 0 By fundamental theorem of calculus ∫ a b f ′ ( x ) d x = f ( b ) − f ( a ) ∫ a b f ′ ( x ) d x = f ( b ) − f ( a ) ∫ 0 b f ′ ( x ) d x = f ( b ) − f ( 0 ) ∫ 0 b f ′ ( x ) d x = − f ( 0 ) [ ∵ f ( b ) = 0 ] Also, d d x ∫ a x f ( t ) d t = f ( x ) d d x ∫ 0 b f ( t ) d t = f ( b ) = 0 h ( x ) = f − 1 ( x ) (Given) π ∫ 0 f ( 0 ) ( h ( x ) ) 2 d x = π ∫ 0 f ( 0 ) ( f − 1 ( x ) ) 2 d x = π ∫ 0 f ( 0 ) ( f − 1 ( x ) ) 2 ⋅ 1 d x

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 6.4, Problem 28E

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To Explain:

For what reason $2 π \int_{0}^{b} x f (x) d x = π \int_{0}^{f (0)} {(h (x))}^{2} d x$ , where h is the inverse of f.

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Chapter 6.4, Problem 27E

Chapter 6.4, Problem 29E

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Chapter 6 Solutions

Calculus - Standalone book

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For what reason 2 π ∫ 0 b x f ( x ) d x = π ∫ 0 f ( 0 ) ( h ( x ) ) 2 d x , where h is the inverse of f .

Solution Summary: The author explains that y = f ( x ) is a decreasing function on [0, b ].

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To Explain:

For what reason $2 π \int_{0}^{b} x f (x) d x = π \int_{0}^{f (0)} {(h (x))}^{2} d x$ , where h is the inverse of f.

Want to see the full answer?

Chapter 6 Solutions

For what reason 2 π ∫ 0 b x f ( x ) d x = π ∫ 0 f ( 0 ) ( h ( x ) ) 2 d x , where h is the inverse of f .

Solution Summary: The author explains that y = f ( x ) is a decreasing function on [0, b ].

Videos

To Explain: For what reason 2π∫0bxf(x)dx =π∫0f(0)(h(x))2dx, where h is the inverse of f.

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Chapter 6 Solutions

To Explain:

For what reason $2 π \int_{0}^{b} x f (x) d x = π \int_{0}^{f (0)} {(h (x))}^{2} d x$ , where h is the inverse of f.