Explanation Given: y 2 + x 2 ≤ R 2 , y ≥ 0 ; δ ( x , y ) = y Formula used: M = ∬ R δ ( x , y ) d A M x = ∬ R y δ ( x , y ) d A M y = ∬ R x δ ( x , y ) d A x ¯ = M y M , y ¯ = M x M Calculation: Here the radius of the semicircle is R so the semicircle cut the x axis at –R and +R, therefore the limit of x will be from –R to +R and the limit of y will be from 0 to R 2 − x 2 . M = ∬ R δ ( x , y ) d A = 2 ∫ 0 R ∫ 0 R 2 − x 2 y d y d x = 2 ∫ 0 R ∫ 0 R 2 − x 2 y 2 2 d y d x = 2 2 ∫ 0 R [ y 2 ] 0 R 2 − x 2 d x = ∫ 0 R ( R 2 − x 2 ) 2 d x = ∫ 0 R ( R 2 − x 2 ) d x = [ R 2 x − x 3 3 ] 0 R = [ R 3 − R 3 3 ] = 2 R 3 3 M x = ∬ R y δ ( x , y ) d A = 2 ∫ 0 R ∫ 0 R 2 − x 2 y . y d y d x = 2 ∫ 0 R ∫ 0 R 2 − x 2 y 2 d y d x = 2 ∫ 0 R ∫ 0 R 2 − x 2 y 3 3 d y d x = 2 3 ∫ 0 R [ y 3 ] 0 R 2 − x 2 d x = 2 3 ∫ 0 R ( R 2 − x 2 ) 3 d x = 2 3 ∫ 0 R ( R 2 − x 2 ) 3 2 d x Put x = R sin θ d x = R cos θ d θ When x = 0 implies θ = 0 and When x = R implies θ = π 2 The integral become 2 3 ∫ 0 π 2 ( R 2 − R 2 sin 2 θ ) 3 2 R cos θ d θ = 2 3 ∫ 0 π 2 [ R 2 ( 1 − sin 2 θ )

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.5, Problem 28E

To determine

To find: The center of mass of the region bounded by $y^{2} + x^{2} \leq R^{2}, y \geq 0;$ $δ (x, y) = y$ .

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Chapter 16.5, Problem 27E

Chapter 16.5, Problem 29E

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Chapter 16 Solutions

CALCULUS (CLOTH)

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The center of mass of the region bounded by y 2 + x 2 ≤ R 2 , y ≥ 0 ; δ ( x , y ) = y .

Solution Summary: The author calculates the coordinates of center of mass of the region bounded by y 2 + x 2 R2,y 0.

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To find: The center of mass of the region bounded by $y^{2} + x^{2} \leq R^{2}, y \geq 0;$ $δ (x, y) = y$ .

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Chapter 16 Solutions

The center of mass of the region bounded by y 2 + x 2 ≤ R 2 , y ≥ 0 ; δ ( x , y ) = y .

Solution Summary: The author calculates the coordinates of center of mass of the region bounded by y 2 + x 2 R2,y 0.

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To find: The center of mass of the region bounded by y2+x2≤R2,y≥0;δ(x,y)=y.

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Chapter 16 Solutions

To find: The center of mass of the region bounded by $y^{2} + x^{2} \leq R^{2}, y \geq 0;$ $δ (x, y) = y$ .