Explanation Given information: Mass density is δ ( x , y ) = 1 and region is lamina bounded by the x − a x e s , y − a x e s , the line x = M and the graph of y = e − x . Formula Used: When mass density is constant then centroid of domain D is ( x ¯ , y ¯ ) , Where x ¯ = 1 A ∬ D x d A , y ¯ = 1 A ∬ D y d A , and A = ∬ D d A Mass of the solid with mass density δ ( x ) and domain D is given by M = ∬ D δ ( x , y ) d A Calculation: As region is bounded by the x = 0 , y = 0 , the line x = M and the graph of y = e − x Thus region is 0 ≤ x ≤ M , 0 ≤ y ≤ e − x Now to evaluate centroid, first find A, which is A = ∬ D d A = ∫ 0 M ( ∫ 0 e − x d y ) d x = ∫ 0 M [ y ] 0 e − x d x = ∫ 0 M ( e − x − 0 ) d x = ∫ 0 M e − x d x = [ − e − x ] 0 M = − e − M − ( e − 0 ) = − e − M + 1 ⇒ A = ( 1 − e − M ) Now, x ¯ = 1 A ∬ D x d A = 1 A ∫ 0 M ( ∫ 0 e − x x d y ) d x = 1 A ∫ 0 M x [ y ] 0 e − x d x = 1 A ∫ 0 M x ( e − x ) d x = 1 A ∫ 0 M x e − x d x To integrate ∫ 0 M x e − x d x , use integration by parts and treat x as first function and e − x as second function Thus, ∫ x e − x d x = x ∫ e − x d x − ∫ ( d d x ( x ) ∫ e − x

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.5, Problem 15E

To determine

Find the centroid of the region where mass density is δ(x,y)=1

Region is: Lamina bounded by the x−axes,y−axes , the line x=M and the graph of y=e−x .

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Chapter 16.5, Problem 14E

Chapter 16.5, Problem 16E

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Chapter 16 Solutions

CALCULUS (CLOTH)

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Find the centroid of the region where mass density is δ ( x , y ) = 1 Region is: Lamina bounded by the x − a x e s , y − a x e s , the line x = M and the graph of y = e − x .

Solution Summary: The author explains the formula for finding the centroid of the region where mass density is delta(x,y)=1

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Find the centroid of the region where mass density is δ(x,y)=1

Region is: Lamina bounded by the x−axes,y−axes , the line x=M and the graph of y=e−x .

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Chapter 16 Solutions

Find the centroid of the region where mass density is δ ( x , y ) = 1 Region is: Lamina bounded by the x − a x e s , y − a x e s , the line x = M and the graph of y = e − x .

Solution Summary: The author explains the formula for finding the centroid of the region where mass density is delta(x,y)=1

Videos

Find the centroid of the region where mass density is δ(x,y)=1 Region is: Lamina bounded by the x−axes,y−axes , the line x=M and the graph of y=e−x .

Want to see the full answer?

Chapter 16 Solutions

Find the centroid of the region where mass density is δ(x,y)=1

Region is: Lamina bounded by the x−axes,y−axes , the line x=M and the graph of y=e−x .