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...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 8.3, Problem 50E

To determine

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 49E

Chapter 8.3, Problem 51E

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8:39 *** TEMU 5G 60% A ferris wheel is 28 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 4 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. What is the amplitude? 14 meters What is the equation of the Midline? y = 16 What is the period? 4 meters minutes The equation that models the height of the ferris wheel after t minutes is: f(t): = ƒ (3) = ·−14(0) + 16 syntax error: you gave an equation, not an expression. syntax error. Check your variables - you might be using an incorrect one. How high are you off of the ground after 3 minutes? Round your answe the nearest meter. ||| <

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Chapter 8 Solutions

CALCULUS (CLOTH)

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.

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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.

Concept explainers

Videos

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.