Explanation Given: y = x 2 − x and y = x − x 3 on [ 0 , 1 ] Calculation: Let’s consider that f and g be continuous f ( x ) ≥ g ( x ) on interval of [ a , b ] Assume that the thin plate of uniform density is ρ The uniform density covers the region “R” between the graphs of y = f ( x ) and y = g ( x ) on the interval of [ a , b ] So, mass of “R” is m = ρ ∫ a b [ f ( x ) − g ( x ) ] d x Centroid of “R” is the point ( x ¯ , y ¯ ) , x ¯ = M y m = ∫ a b x [ f ( x ) − g ( x ) ] d x ∫ a b [ f ( x ) − g ( x ) ] d x And, y ¯ = M x m = 1 2 ∫ a b { [ f ( x ) ] 2 − [ g ( x ) ] 2 } d x ∫ a b [ f ( x ) − g ( x ) ] d x Here, f ( x ) = x − x 3 and g ( x ) = x 2 − x f ( x ) ≥ g ( x ) in the interval of [ 0 , 1 ] m = ρ ∫ a b [ f ( x ) − g ( x ) ] d x = ρ ∫ 0 1 [ x − x 3 − ( x 2 − x ) ] d x = ρ ∫ 0 1 [ 2 x − x 3 − x 2 ] d x With proceed further, the solution will be, = ρ [ x 2 − x 4 4 − x 3 3 ] x

Student's Solution and Survival Manual for Calculus

7th Edition

ISBN: 9781524934040

Author: STRAUSS MONTY J, TODA MAGDALENA DANIELE, SMITH KARL J

Publisher: Kendall Hunt Publishing

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Chapter 6, Problem 28PE

To determine

To calculate: the centroid of the homogenous region

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Chapter 6, Problem 27PE

Chapter 6, Problem 29PE

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Chapter 6 Solutions

Student's Solution and Survival Manual for Calculus

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the centroid of the homogenous region

Solution Summary: The author calculates the centroid of the homogenous region between y=x2-x and

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Chapter 6 Solutions