Explanation Given: Region bounded by y = 1 − x 2 and y = 0 . 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 y-axis. So, x C M = 0 First, evaluate the total mass and we get: M = ∬ D δ ( x , y ) d A = ∫ − 1 1 ∫ 0 1 − x 2 ( 1 ) d y d x = ∫ − 1 1 [ y ] 0 1 − x 2 d x = ∫ − 1 1 ( 1 − x 2 ) d x = 2 [ x − x 3 3 ] − 1 1 = [ ( 1 − 1 3 ) − ( − 1 − ( − 1 ) 3 3 ) ] = [ ( 2 3 ) − ( − 2 3 ) ] = 4 3 Therefore, the total mass is: M = 4 3 Now, evaluate M x =

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.5, Problem 11E

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To calculate:

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

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Chapter 16.5, Problem 10E

Chapter 16.5, Problem 12E

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Chapter 16 Solutions

CALCULUS (CLOTH)

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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 $δ (x, y) = 1$ .