Explanation Given: Let a > 0 and the function is y = a x − a x 2 . Formula used: Volume V = ∫ a b A ( x ) d x Where a ≤ x ≤ b and A = π [ ( Outer radius ) 2 − ( Inner radius ) 2 ] . Calculation: Let a > 0 and the function is y = a x − a x 2 . Set a x − a x 2 = 0 . Solve the above expression. a x − a x 2 = 0 x − a x 2 = 0 x ( 1 − a x ) = 0 That is, x = 0 or ( 1 − a x ) = 0 x = 0 or x = 1 a Use the washer method, cross sections are circular disks with radius R = a x − a x 2 . Now, find the volume of the solid. Volume = π ∫ 0 1 a a 2 ( x − a x 2 ) d x = π ∫ 0 1 a a 2 x − a 3 x 2 d x Solve the integral...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 6, Problem 47CRE

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Show that the volume obtained when the region between $y = a \sqrt{x - a x^{2}}$ and the $x - axis$ is rotated about the $x - axis$ is independent of the constant $a$ .

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Chapter 6, Problem 46CRE

Chapter 6, Problem 48CRE

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Chapter 6 Solutions

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Show that the volume obtained when the region between y = a x − a x 2 and the x − axis is rotated about the x − axis is independent of the constant a .

Solution Summary: The author explains how the volume is independent of the constant a.

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Show that the volume obtained when the region between $y = a \sqrt{x - a x^{2}}$ and the $x - axis$ is rotated about the $x - axis$ is independent of the constant $a$ .

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Show that the volume obtained when the region between y = a x − a x 2 and the x − axis is rotated about the x − axis is independent of the constant a .

Solution Summary: The author explains how the volume is independent of the constant a.

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Show that the volume obtained when the region between y=ax−ax2 and the x−axis is rotated about the x−axis is independent of the constant a.

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Chapter 6 Solutions

Show that the volume obtained when the region between $y = a \sqrt{x - a x^{2}}$ and the $x - axis$ is rotated about the $x - axis$ is independent of the constant $a$ .