Explanation Given: The density function is constant. The region isantetrahedronwhich is bounded by the coordinate planes and x + y 2 + z 3 = 1 . Formula used: The mass is given by ∭ E ρ ( x , y , z ) d A . The center of the mass is given by ( x ¯ , y ¯ , z ¯ ) , where x ¯ = 1 m ∭ E x ρ ( x , y , z ) d V , y ¯ = 1 m ∭ E y ρ ( x , y , z ) d V and z ¯ = 1 m ∭ E z ρ ( x , y , z ) d V . Calculation: From the given region, it is observed that the height of the tetrahedron is given by 3. Since the density is constant, the volume of the surface is equal to the mass. Thus, m = 1 3 ( 3 ) ( 1 ) = 1 The x- coordinate of the center of mass is given by, x ¯ = 1 1 ∫ 0 1 ∫ 0 2 − 2 x ∫ 0 3 ( 1 − x − y 2 ) x d z d y d x = ∫ 0 1 ∫ 0 2 − 2 x x [ z ] 0 3 ( 1 − x − y 2 ) d y d x = ∫ 0 1 ∫ 0 2 − 2 x x [ 3 ( 1 − x − y 2 ) − 0 ] d y d x = 3 ∫ 0 1 ∫ 0 2 − 2 x ( x − x 2 − x y 2 ) d y d x Integrate with respect to y and apply the limit. x ¯ = 3 ∫ 0 1 [ ( x − x 2 ) y − x y 2 2 ( 2 ) ] 0 2 − 2 x d x = 3 ∫ 0 1 [ ( ( x − x 2 ) ( 2 − 2 x ) − x ( 2 − 2 x ) 2 4 ) − ( ( x − x 2 ) ( 0 ) − x ( 0 ) 2 4 ) ] d x = 3 ∫ 0 1 [ ( ( 2 x − 2 x 2 − 2 x 2 + 2 x 3 ) − ( 4 x − 8 x 2 + 4 x 3 ) 4 ) − ( 0 − 0 ) ] d x = 3 ∫ 0 1 ( 2 x − 4 x 2 + 2 x 3 − x + 2 x 2 − x 3 ) d x = 3 ∫ 0 1 ( x − 2 x 2 + x 3 ) d x Integrate with respect to x and apply the limit. x ¯ = 3 [ x 2 2 − 2 x 3 3 + x 4 4 ] 0 1 = 3 [ ( 1 2 2 − 2 ( 1 ) 3 3 + 1 4 4 ) − ( 0 2 2 − 2 ( 0 ) 3 3 + 0 4 4 ) ] = 3 [ ( 1 2 − 2 3 + 1 4 ) − ( 0 − 0 + 0 ) ] = 3 ( 1 12 ) = 1 4 The y- coordinate of the center of mass is given by, y ¯ = 1 1 ∫ 0 1 ∫ 0 2 − 2 x ∫ 0 3 ( 1 − x − y 2 ) y d z d y d x = ∫ 0 1 ∫ 0 2 − 2 x y [ z ] 0 3 ( 1 − x − y 2 ) d y d x = ∫ 0 1 ∫ 0 2 − 2 x y [ 3 ( 1 − x − y 2 ) − 0 ] d y d x = 3 ∫ 0 1 ∫ 0 2 − 2 x ( y − x y − y 2 2 ) d y d x Integrate with respect to y and apply the limit

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 13, Problem 62RE

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Chapter 13, Problem 61RE

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Chapter 13 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 13.1 - 1. Write an iterated integral that gives the...Ch. 13.1 - Write an iterated integral that gives the volume...Ch. 13.1 - Write two iterated integrals that equal Rf(x,y)dA,...Ch. 13.1 - Consider the integral 1311(2y2+xy)dydx. State the...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...

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The coordinates of the center of the mass.

Solution Summary: The author explains that the density function is constant and the region isantetrahedron is bounded by coordinate planes.

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Chapter 13 Solutions