Explanation Given: Quarter circle x 2 + y 2 ≤ R 2 , x ≥ 0 , y ≥ 0 and Density δ ( x , y ) = 1 . Formula Used: Center of mass is given by: ( x C M , y C M ) = ( M y M , M x M ) ... ( 1 ) M = ∬ D δ ( x , y ) d A M y = ∬ D x δ ( x , y ) d A and M x = ∬ D y δ ( x , y ) d A Calculation: Here, assume the region in polar coordinate x = r cos θ , y = r sin θ and 0 ≤ r ≤ R , 0 ≤ θ ≤ π 2 . First, evaluate the total mass and we get: M = ∬ D δ ( x , y ) d A = ∫ θ = 0 π 2 ∫ r = 0 R ( 1 ) r d r d θ = ∫ 0 π 2 [ r 2 2 ] 0 R d θ = ∫ 0 π 2 R 2 2 d θ = R 2 2 [ θ ] 0 π 2 = π R 2 4 Therefore, the total mass is: M = π R 2 4 Now, evaluate M y = ∬ D x δ ( x , y ) d A and we get: M y = ∬ D x δ ( x , y ) d A = ∫ θ = 0 π 2 ∫ r = 0 R ( r cos θ ) ( 1 ) r d r d θ = ∫

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.5, Problem 13E

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The centroid of the given region assuming the density $δ (x, y) = 1$ .

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Chapter 16.5, Problem 12E

Chapter 16.5, Problem 14E

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The centroid of the given region assuming the density δ ( x , y ) = 1 .

Solution Summary: The author calculates the centroid of the given region assuming the density (x, y ) = 1.

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The centroid of the given region assuming the density $δ (x, y) = 1$ .

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Chapter 16 Solutions

The centroid of the given region assuming the density δ ( x , y ) = 1 .

Solution Summary: The author calculates the centroid of the given region assuming the density (x, y ) = 1.

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To calculate: The centroid of the given region assuming the density δ(x,y)=1.

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Chapter 16 Solutions

To calculate:

The centroid of the given region assuming the density $δ (x, y) = 1$ .