Explanation Given: Let D be the domain between the line y = b x a and the parabola y = b x 2 a 2 ,where a , b > 0 . Assume mass density is δ ( x , y ) = 1 . Calculation: Centroid ℝ 2 is given by: ( x ¯ , y ¯ ) = ( 1 A ∬ D x d A , 1 A ∬ D y d A ) ... ( 1 ) Now, the area will be: A = ∬ D x y d A = ∫ x = 0 a ∫ y = b x 2 a 2 b x a x y d y d x = ∫ 0 a x [ y 2 2 ] b x 2 a 2 b x a d x = 1 2 ∫ 0 a x [ ( b x a ) 2 − ( b x 2 a 2 ) 2 ] d x = 1 2 ∫ 0 a [ b 2 x 3 a 2 − b 2 x 5 a 4 ] d x = b 2 8 a 2 [ x 4 ] 0 a − b 2 12 a 4 [ x 6 ] 0 a = a 2 b 2 8 − a 2 b 2 12 = a 2 b 2 24 Therefore, the area is: A = a 2 b 2 24 Now, calculate 1 A ∬ D x δ ( x , y ) d A and we get: 1 A ∬ D x d A = 24 a 2 b 2 ∫ x = 0 a ∫ y = b x 2 a 2 b x a x δ ( x , y ) d A = 24 a 2 b 2 ∫ x = 0 a ∫ y = b x 2 a 2 b x a x ( 1 ) d x d y = 24 a 2 b 2 ∫ 0 a x d x ∫ b x 2 a 2 b x a d y = 24 a 2

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.5, Problem 39E

To determine

To calculate:

Centroid of the region.

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Chapter 16.5, Problem 38E

Chapter 16.5, Problem 40E

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Chapter 16 Solutions

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Centroid of the region.

Solution Summary: The author calculates the centroid of the region. Let D be the domain between the line y = b x a and the parabola

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Chapter 16 Solutions

To calculate:

Centroid of the region.