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 infinite bounded by the x = 0 , y = 0 and the graph of y = e − x Thus region is 0 ≤ x ≤ ∞ , 0 ≤ y ≤ e − x Now to evaluate centroid, first find A, which is A = ∬ D d A = ∫ 0 ∞ ( ∫ 0 e − x d y ) d x = ∫ 0 ∞ [ y ] 0 e − x d x = ∫ 0 ∞ ( e − x − 0 ) d x = ∫ 0 ∞ e − x d x = [ − e − x ] 0 ∞ = − e − ∞ − ( e − 0 ) = 0 + 1 = 1 ⇒ A = 1 Now, x ¯ = 1 A ∬ D x d A = 1 1 ∫ 0 ∞ ( ∫ 0 e − x x d y ) d x = ∫ 0 ∞ x [ y ] 0 e − x d x = ∫ 0 ∞ x ( e − x ) d x = ∫

4th Edition

ISBN: 9781319309671

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 15.5, Problem 16E

To determine

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

Region is: Infinite lamina bounded by the x−axes,y−axes, the graph of y=e−x.

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Chapter 15.5, Problem 15E

Chapter 15.5, Problem 17E

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Chapter 15 Solutions

CALCULUS: EARLY TRANS 4TH ED W/ ACCESS

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

Solution Summary: The author explains how to calculate 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: Infinite lamina bounded by the x−axes,y−axes, the graph of y=e−x.

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Chapter 15 Solutions

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

Solution Summary: The author explains how to calculate 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: Infinite lamina bounded by the x−axes,y−axes, the graph of y=e−x.

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Chapter 15 Solutions

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

Region is: Infinite lamina bounded by the x−axes,y−axes, the graph of y=e−x.