Explanation Given information: The given function is f ( x ) = 9 − x 2 and the interval is [ 0 , 3 ] . 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 ) = 9 − x 2 Consider the given interval. [ 0 , 3 ] Here, a = 0 , b = 3 Centroid does not depend on density. So, we can consider ρ = 1 . So, the total mass M can be calculated as follows: M = ∫ 0 3 ( 9 − x 2 ) d x = [ 9 x − x 3 3 ] 0 3 = [ 27 − 9 ] = 18 M x can be calculated as shown below

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 8.4, Problem 17E

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The centroid of the region lying underneath the graph of the given function over the interval [0,3].

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Chapter 8.4, Problem 16E

Chapter 8.4, Problem 18E

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Chapter 8 Solutions

Calculus: Early Transcendentals

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The centroid of the region lying underneath the graph of the given function over the interval [ 0 , 3 ] .

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:

The centroid of the region lying underneath the graph of the given function over the interval [0,3].

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Chapter 8 Solutions

The centroid of the region lying underneath the graph of the given function over the interval [ 0 , 3 ] .

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: The centroid of the region lying underneath the graph of the given function over the interval [0,3].

Want to see the full answer?

Chapter 8 Solutions

To find:

The centroid of the region lying underneath the graph of the given function over the interval [0,3].