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: 9780321965165

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

Publisher: PEARSON

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Chapter 13, Problem 62RE

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To find: The coordinates of the center of the mass.

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Chapter 13, Problem 61RE

Chapter 13, Problem 63RE

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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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To find: The coordinates of the center of the mass.

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Chapter 13 Solutions