Explanation Given: The given function is f ( x ) = ln x , [ 1 , 2 ] Formula Used: The moment with respect to the y-axis M y , can be calculated by M y = δ ∫ a b x f ( x ) d x The moment with respect to x-axis M x , can be calculated by M x = 1 2 δ ∫ a b ( f ( x ) ) 2 d x Total mass M can be calculated by M = δ ∫ a b f ( x ) d x The center of mass coordinates are the moments divided by the total mass: x ¯ = M y M a n d y ¯ = M x M Calculation: The given function is f ( x ) = ln x The given interval is [ 1 , 2 ] So, a = 1 a n d b = 2 The centroid does not depend on δ , so set δ = 1 . First, we have to calculate the moment with respect to x-axis. Substitute the value of a , b , δ a n d f ( x ) in an equation M x = 1 2 δ ∫ a b ( f ( x ) ) 2 d x , we get M x = 1 2 ∫ 1 2 ( ln x ) 2 d x = 1 2 | ( ln x ) 2 x − ∫ 2 ln x d x | 1 2 = 1 2 | x ln 2 x − 2 ( x ln ( x ) − x ) | 1 2 = 1 2 ( ( 2 ln 2 2 − 2 ( 2 ln ( 2 ) − 2 ) ) − ( ln 2 1 − 2 ( ln ( 1 ) − 1 ) ) ) = ( ln ( 2 ) − 1 ) 2 Now, we need to calculate the moment with respect to y-axis Substitute the value of a , b , δ a n d f ( x ) in an equation M y = δ ∫ a b x f ( x ) d x , we get M y = ∫ 1 2 x ( ln ( x ) ) d x = | ln ( x

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 9.4, Problem 20E

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To find the centroid of the region lying underneath the graph of the function over the given interval.

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Chapter 9.4, Problem 19E

Chapter 9.4, Problem 21E

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Chapter 9 Solutions

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To find the centroid of the region lying underneath the graph of the function over the given interval.

Solution Summary: The author explains how to determine the centroid of the region lying underneath the graph of a function over the given interval.

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To find the centroid of the region lying underneath the graph of the function over the given interval.

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Chapter 9 Solutions