Explanation Given information: The given function is f ( x ) = sin x and the interval is [ 0 , π ] . Formula used: Centroid of the region lying underneath the graph of a function y = f ( x ) over an interval [ a , b ] , is given by ( x ¯ , y ¯ ) = ( M y M , M x M ) Where, M , M x , M y can be calculated by the following formula. M = ρ ∫ a b f ( x ) d x M x = 1 2 ρ ∫ a b [ f ( x ) ] 2 d x M y = ρ ∫ a b x f ( x ) d x The following definite integral can be calculated as shown below. ∫ sin x d x = − cos x + C ∫ x sin x d x = − x cos x + sin x + C ∫ ( sin x ) 2 d x = − 1 2 ( x − sin 2 x 2 ) + C Calculation: Consider the given function. f ( x ) = sin x Consider the given interval. [ 0 , π ] Here, a = 0 , b = π Centroid does not depend on density. So, we can consider ρ = 1 . So, the total mass M can be calculated as follows: M = ∫ 0 π sin x d x = [ − cos x ] 0 π = − [ cos π − cos 0 ] = − [ − 1 − 1 ] = 2 M x can be calculated as shown below

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

4th Edition

ISBN: 9781319312886

Author: Rogawski, Jon, Adams, Colin, FRANZOSA, Robert

Publisher: W. H. Freeman

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Chapter 8.4, Problem 21E

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Thecentroid of the region lying underneath the graph of the given function over the interval [0,π].

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Chapter 8.4, Problem 20E

Chapter 8.4, Problem 22E

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Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

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Thecentroid of the region lying underneath the graph of the given function over the interval [ 0 , π ] .

Solution Summary: The author calculates the centroid of the region lying underneath the graph of a function y=f(x) over an interval.

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To find:

Thecentroid of the region lying underneath the graph of the given function over the interval [0,π].

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Chapter 8 Solutions

Thecentroid of the region lying underneath the graph of the given function over the interval [ 0 , π ] .

Solution Summary: The author calculates the centroid of the region lying underneath the graph of a function y=f(x) over an interval.

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To find: Thecentroid of the region lying underneath the graph of the given function over the interval [0,π].

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Chapter 8 Solutions

To find:

Thecentroid of the region lying underneath the graph of the given function over the interval [0,π].