Explanation Given information: The given function is f ( x ) = x and the interval is [ 1 , 4 ] . 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 power rule of indefinite integrals. ∫ x n d x = x n n + 1 + C Calculation: Consider the given function. f ( x ) = x Consider the given interval. [ 1 , 4 ] Here, a = 1 , b = 4 Centroid does not depend on density. So, we can consider ρ = 1 . So, the total mass M can be calculated as follows: M = ∫ 1 4 x d x = [ 2 x 3 / 2 3 ] 1 4 = [ 2 ( 4 ) 3 / 2 3 − 2 ( 1 ) 3 / 2 3 ] = 16 3 − 2 3 = 14 3 M x can be calculated as shown below

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ISBN: 9781319308858

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 8.4, Problem 15E

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Thecentroid of the region lying underneath the graph of the given function over the interval [1,4].

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Chapter 8.4, Problem 14E

Chapter 8.4, Problem 16E

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Thecentroid of the region lying underneath the graph of the given function over the interval [ 1 , 4 ] .

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 [1,4].

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Chapter 8 Solutions

Thecentroid of the region lying underneath the graph of the given function over the interval [ 1 , 4 ] .

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 [1,4].

Want to see the full answer?

Chapter 8 Solutions

To find:

Thecentroid of the region lying underneath the graph of the given function over the interval [1,4].