Explanation Given : The semicircle x 2 + y 2 ≤ R 2 , x ≥ 0 Calculation : Step 1: The average value of the x-coordinates of a point on the semicircle D is x ¯ = 1 A r e a ( D ) ∫ ∫ D x d A Step 2: The area of semicircle is π R 2 2 To compute the double integral, we identify the inequalities defining D as a horizontally simple region: − R ≤ y ≤ R , 0 ≤ x ≤ R 2 − y 2 Step 3: Therefore, the average value of the x-coordinate is x ¯ = 1 π R 2 2 ∫ − R R ∫ 0 R 2 − y 2 x d x d y = 2 π R 2 ∫ − R R x 2 2 | 0 R 2 − y 2 d y = 2 π R 2 ∫ − R R ( R 2 − y 2 ) 2 − 0 2 2 d y = 1 π R 2 ∫ − R R ( R 2 − y 2 ) d y = 2 π R 2 ∫ 0 R ( R 2 − y 2 ) d y = 2 π R 2 ( R 2 y − y 3 3 ) | y = 0 R = 2 π R 2 ( R 3 − R 3 3 ) = 2 π R 2 ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.2, Problem 54E

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The average value of the x-coordinate of a point on the semicircle $x^{2} + y^{2} \leq R^{2}, x \geq 0$ and the average value of the y-coordinate

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Chapter 16.2, Problem 53E

Chapter 16.2, Problem 55E

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Chapter 16 Solutions

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The average value of the x-coordinate of a point on the semicircle x 2 + y 2 ≤ R 2 , x ≥ 0 and the average value of the y-coordinate

Solution Summary: The author calculates the double integral by identifying the inequalities defining D as a horizontally simple region.

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The average value of the x-coordinate of a point on the semicircle $x^{2} + y^{2} \leq R^{2}, x \geq 0$ and the average value of the y-coordinate

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Chapter 16 Solutions

The average value of the x-coordinate of a point on the semicircle x 2 + y 2 ≤ R 2 , x ≥ 0 and the average value of the y-coordinate

Solution Summary: The author calculates the double integral by identifying the inequalities defining D as a horizontally simple region.

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The average value of the x-coordinate of a point on the semicircle x2+y2≤R2,x≥0 and the average value of the y-coordinate

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Chapter 16 Solutions

The average value of the x-coordinate of a point on the semicircle $x^{2} + y^{2} \leq R^{2}, x \geq 0$ and the average value of the y-coordinate