a. Explanation Given information: The circles have radii a and b . The circles have their centers at the origin. Calculation: The plate is symmetric about x -axis. Therefore, the centroid y ¯ = 0 . On the region [ 0 , a ] , Consider a small vertical strip with the center of mass of ( x ˜ , y ˜ ) = ( x , b 2 − x 2 + a 2 − x 2 2 ) , length b 2 − x 2 − a 2 − x 2 , width d x , and area ( b 2 − x 2 − a 2 − x 2 ) d x . Mass of the small vertical strip is d m = δ d A . For d A = ( b 2 − x 2 − a 2 − x 2 ) d x . d m = δ ( b 2 − x 2 − a 2 − x 2 ) d x On the region [ a , b ] , Consider a small vertical strip with the center of mass of ( x ˜ , y ˜ ) = ( x , b 2 − x 2 2 ) , length b 2 − x 2 , width d x , and area b 2 − x 2 d x . Mass of the small vertical strip is d m = δ d A . For d A = b 2 − x 2 d x , d m = δ b 2 − x 2 d x Find the moment of the strip about y -axis using the relation: M y = ∫ x ˜ d m Substitute δ ( b 2 − x 2 − a 2 − x 2 ) d x for d m . M y = ∫ 0 a x δ ( b 2 − x 2 − a 2 − x 2 ) d x + ∫ a b x δ b 2 − x 2 d x = δ ∫ 0 a x ( b 2 − x 2 ) 1 / 2 d x − δ ∫ 0 a x ( a 2 − x 2 ) 1 / 2 d x + δ ∫ a b x ( b 2 − x 2 ) 1 / 2 d x = − δ 2 [ 2 ( b 2 − x 2 ) 3 / 2 3 ] 0 a + δ 2 [ 2 ( a 2 − x 2 ) 3 / 2 3 ] 0 a − δ 2 [ 2 ( b 2 − x 2 ) 3 / 2 3 ] a b = − δ 3 [ ( b 2 − a 2 ) 3 / 2 − ( b 2 ) 3 / 2 ] + δ 3 [ 0 − ( a 2 ) 3 / 2 ] − δ 3 [ 0 − ( b 2 − a 2 ) 3 / 2 ] M y = δ b 3 3 − δ a 3 3 = δ ( b 3 − a 3 ) 3 Find the moment of the strip about x -axis using the relation: M x = ∫ y ˜ d m Substitute δ ( b 2 − x 2 − a 2 − x 2 ) d x for d m b. To determine Obtain the coordinate limits of the centroid if a → b .

Thomas' Calculus: Early Transcendentals (14th Edition)

14th Edition

ISBN: 9780134439020

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 6, Problem 15AAE

To determine

Calculate the centroid of the region bounded by two concentric circles and the coordinate axes in the first quadrant.

To determine

Obtain the coordinate limits of the centroid if a→b.

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Chapter 6, Problem 14AAE

Chapter 6, Problem 16AAE

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