Explanation Given information: R is a region in the plane, R can be partitioned into two sub-regions R 1 and R 2 . ( x 1 ¯ , y 1 ¯ ) and ( x 2 ¯ , y 2 ¯ ) be the centroid of R 1 and R 2 , A 1 and A 2 be the areas of the sub-regions. Formula used: The area of R is given by A = ρ ∫ a b ( f ( x ) − g ( x ) ) d x And the centroid of R is the point ( x ¯ , y ¯ ) such that x ¯ = M y A = ∫ a b x [ f ( x ) − g ( x ) ] d x ∫ a b [ f ( x ) − g ( x ) ] d x a n d y ¯ = M x A = 1 2 ∫ a b [ [ f ( x ) ] 2 − [ g ( x ) ] 2 ] d x ∫ a b [ f ( x ) − g ( x ) ] d x Proof: Since ( x 1 ¯ , y 1 ¯ ) and A 1 are the centroid and area of the region R 1 . ⇒ x 1 ¯ = M y 1 A 1 ⇒ M y 1 = A 1 x 1 ¯ y 1 ¯ = M x 1 A 1 ⇒ M x 1 = A 1 y 1 ¯ Since ( x 2 ¯ , y 2 ¯ ) b and A 2 are the centroid and area of the region R 2 . ⇒ x 2 ¯ = M y 2 A 2 ⇒ M y 2 = A 2 x 2 ¯ y 2 ¯ = M x 2 A 2 ⇒ M x 2 = A 2 y 2 ¯ ( x ¯ , y ¯ ) is the centroid of R

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 6.5, Problem 59PS

To determine

To prove: (x¯,y¯) is the centroid of R, where x¯=A1x1+A2x2A1+A2 and y¯=A1y1+A2y2A1+A2

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Chapter 6.5, Problem 58PS

Chapter 6.5, Problem 60PS

Chapter 6 Solutions

Calculus

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To prove : ( x ¯ , y ¯ ) is the centroid of R, where x ¯ = A 1 x 1 + A 2 x 2 A 1 + A 2 and y ¯ = A 1 y 1 + A 2 y 2 A 1 + A 2

Solution Summary: The author explains that R is a region in the plane, and can be partitioned into two sub-regions.

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To prove: (x¯,y¯) is the centroid of R, where x¯=A1x1+A2x2A1+A2 and y¯=A1y1+A2y2A1+A2

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Chapter 6 Solutions