Explanation Given: Region bounded by y 2 = x + 4 and x = 4 . 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 ) 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, region is symmetric with respect to x-axis. So, y C M = 0 It is given that x = 4 and y 2 = x + 4 , so y 2 = 8 y = ± 2 2 First, evaluate the total mass and we get: M = ∬ D δ ( x , y ) d A = ∫ − 2 2 2 2 ∫ y 2 − 4 4 ( 1 ) d x d y = ∫ − 2 2 2 2 [ x ] y 2 − 4 4 d y = ∫ − 2 2 2 2 ( 4 − y 2 + 4 ) d y = ∫ − 2 2 2 2 ( 8 − y 2 ) d y = [ 8 y − y 3 3 ] − 2 2 2 2 M = [ ( 16 2 − 16 2 3 ) − ( − 16 2 − − 16 2 3 ) ] = [ ( 32 2 3 ) − ( − 32 2 3 ) ] = 64 2 3 Therefore, the total mass is:

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ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

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Chapter 16.5, Problem 12E

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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 11E

Chapter 16.5, Problem 13E

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Chapter 16 Solutions

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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 $δ (x, y) = 1$ .