Explanation Given: The function is y = 1 − x 2 . Key concept used: Formula of volume: Volume = 2 π ∫ 0 1 x y d x Steps of integration by substitution: If the integral is in the below form: ∫ f ( g ( x ) ) g ′ ( x ) d x Let u = g ( x ) and d u = g ′ ( x ) d x The intgeral becomes, ∫ f ( u ) d u . Thus, integrate f ( u ) . After that substitute g ( x ) back. Integration formula: ∫ cos x d x = sin x + c , ∫ 1 d x = x + c Calculation: The function is y = x 1 − x 2 . Find the volume. Volume = 2 π ∫ 0 1 x ( x 1 − x 2 ) d x = 2 π ∫ 0 1 x 2 1 − x 2 d x (1) Let x = sin θ . Differentiate x = sin θ . d x = cos θ d θ Substitute sin θ for x and cos θ d θ for d x in equation (2). 2 π ∫ x 2 1 − x 2 d x = 2 π ∫ sin 2 θ 1 − ( sin θ ) 2 ⋅ cos θ d θ = 2 π ∫ sin 2 θ cos θ cos θ d θ = 2 π ∫ ( 1 − cos 2 θ ) cos 2 θ d θ = 2 π ∫ ( cos 2 θ − cos 4 θ ) d θ Integrate the function. 2 π ∫ ( cos 2 θ − cos 4 θ ) d θ = 2 π [ ∫ cos 2 θ d θ − ∫ cos 4 θ d θ ] = 2 π [ ∫ cos 2 θ d θ − cos 3 θ sin θ 4 − 3 4 ∫ cos 2 θ d θ ] = 2 π [ − cos 3 θ sin θ 4 + 1 4 ∫ cos 2 θ d θ ] Integrate the remaining part...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.3, Problem 48E

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Find the volume of the solid obtained by revolving the graph of $y = x \sqrt{1 - x^{2}}$ over $[0, 1]$ about the $y -$ axis.

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Chapter 8.3, Problem 47E

Chapter 8.3, Problem 49E

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Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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Find the volume of the solid obtained by revolving the graph of y = x 1 − x 2 over [ 0 , 1 ] about the y − axis.

Solution Summary: The author explains how to determine the volume of the function y = 1 x 2.

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Find the volume of the solid obtained by revolving the graph of $y = x \sqrt{1 - x^{2}}$ over $[0, 1]$ about the $y -$ axis.

Want to see the full answer?

Chapter 8 Solutions

Find the volume of the solid obtained by revolving the graph of y = x 1 − x 2 over [ 0 , 1 ] about the y − axis.

Solution Summary: The author explains how to determine the volume of the function y = 1 x 2.

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Find the volume of the solid obtained by revolving the graph of y=x1−x2 over [ 0,1 ] about the y− axis.

Want to see the full answer?

Chapter 8 Solutions

Find the volume of the solid obtained by revolving the graph of $y = x \sqrt{1 - x^{2}}$ over $[0, 1]$ about the $y -$ axis.