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

Calculus - Standalone book

Ch. 6.1 - Prob. 1PQ Ch. 6.1 - Prob. 2PQ Ch. 6.1 - Prob. 3PQ Ch. 6.1 - Prob. 4PQ Ch. 6.1 - Prob. 5PQ Ch. 6.1 - Prob. 6PQ Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E

Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E Ch. 6.1 - Prob. 11E Ch. 6.1 - Prob. 12E Ch. 6.1 - Prob. 13E Ch. 6.1 - Prob. 14E Ch. 6.1 - Prob. 15E Ch. 6.1 - Prob. 16E Ch. 6.1 - Prob. 17E Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - Prob. 26E Ch. 6.1 - Prob. 27E Ch. 6.1 - Prob. 28E Ch. 6.1 - Prob. 29E Ch. 6.1 - Prob. 30E Ch. 6.1 - Prob. 31E Ch. 6.1 - Prob. 32E Ch. 6.1 - Prob. 33E Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - Prob. 38E Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.1 - Prob. 41E Ch. 6.1 - Prob. 42E Ch. 6.1 - Prob. 43E Ch. 6.1 - Prob. 44E Ch. 6.1 - Prob. 45E Ch. 6.1 - Prob. 46E Ch. 6.1 - Prob. 47E Ch. 6.1 - Prob. 48E Ch. 6.1 - Prob. 49E Ch. 6.1 - Prob. 50E Ch. 6.1 - Prob. 51E Ch. 6.1 - Prob. 52E Ch. 6.1 - Prob. 53E Ch. 6.1 - Prob. 54E Ch. 6.1 - 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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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Chapter 6 Solutions

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 $a$ .