Explanation Given information: A region between y = x + 4 and y = 2 − x for 0 ≤ x ≤ 2 line y = 3. Formula used: The x -moment of a lamina S of density ρ occupying the region between the graphs of x = g 1 ( y ) and x = g 2 ( y ) over an interval [ a , b ] with g 1 ( y ) > g 2 ( y ) , is given by M x = ρ ∫ a b y ( g 1 ( y ) − g 2 ( y ) ) d y The y -moment of a lamina S of density ρ occupying the region between the graphs of y = f 1 ( x ) and y = f 2 ( x ) over an interval [ c , d ] with f 1 ( x ) > f 2 ( x ) is given by M y = ρ ∫ c d x ( f 1 ( x ) − f 2 ( x ) ) d x The area of a horizontal lamina S of density ρ occupying the region between the graphs of x = g 1 ( y ) and x = g 2 ( y ) over an interval [ a , b ] with g 1 ( y ) > g 2 ( y ) , is given by A = ∫ a b ( g 1 ( y ) − g 2 ( y ) ) d y The area of a vertical lamina S of density ρ occupying the region between the graphs of y = f 1 ( x ) and y = f 2 ( x ) over an interval [ c , d ] with f 1 ( x ) > f 2 ( x ) is given by A = ∫ c d ( f 1 ( x ) − f 2 ( x ) ) d x Centroid of a lamina S 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 = ρ A Calculation: A sketch of the region shown below The region is clearly symmetric about the line y = 3. Therefore, centroid of the region lies on the line y = 3. 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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Question

Chapter 8.4, Problem 23E

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Sketch the region between y=x+4 and y=2−x for 0≤x≤2 line y=3. Compute the moments and centroid of the region.

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Chapter 8.4, Problem 22E

Chapter 8.4, Problem 24E

Chapter 8 Solutions

Calculus: Early Transcendentals

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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Sketch the region between y = x + 4 and y = 2 − x for 0 ≤ x ≤ 2 line y = 3. Compute the moments and centroid of the region.

Solution Summary: The author calculates the moments and centroid of the region. The x -moment of a lamina S of density is given by

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Sketch the region between y=x+4 and y=2−x for 0≤x≤2 line y=3. Compute the moments and centroid of the region.

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