Explanation Given: ρ ( x , y , z ) = k , Q : 2 x + 3 y + 6 z = 12 , x = 0 , y = 0 , z = 0 Formula Used: Total mass of the solid region Q = ∭ Q ρ ( x , y , z ) d v M y z = ∭ Q x ρ ( x , y , z ) d v Calculation: Total mass of the solid region Q = ∭ Q ρ ( x , y , z ) d v ⇒ Q = ∭ Q K d v = K ∫ 0 6 ∫ 0 4 − ( 2 x / 3 ) ∫ 0 2 − ( y / 2 ) − ( x / 3 ) d z d y d x = K ∫ 0 6 ∫ 0 4 − ( 2 x / 3 ) [ z ] 0 2 − ( y / 2 ) − ( x / 3 ) d y d x = K ∫ 0 6 ∫ 0 4 − ( 2 x / 3 ) [ z ] 0 2 − ( y / 2 ) − ( x / 3 ) d y d x = K ∫ 0 6 ∫ 0 4 − ( 2 x / 3 ) ( 2 − y 2 − x 3 ) d y d x = K ∫ 0 6 [ ( 2 − x 3 ) y − y 2 4 ] 0 4 − ( 2 x / 3 ) d x = K ∫ 0 6 [ ( 2 − x 3 ) ( 4 − ( 2 x / 3 ) ) − ( 4 − ( 2 x / 3 ) ) 2 4 ] d x = k ∫ 0 6 [ 8 − 4 x 3 − 4 x 3 + 2 x 2 9 − 1 4 ( 16 + 4 x 2 9 − 16 x 3 ) ] d x = k [ 8 x − 4 x 2 3 + 2 x 3 27 − 1 4 ( 16 x + 4 x 3 27 − 8 x 2 3 ) ] 0 6 = k ( 48 − 48 + 16 − 1 4 ( 96 + 32 − 96 ) ) = 8 k The first moment about y z plane is M y z = ∭ Q x ρ ( x , y , z

Calculus: Early Transcendental Functions (MindTap Course List)

6th Edition

ISBN: 9781285774770

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 14.6, Problem 37E

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To Calculate: The mass and indicated coordinate of the centre of the mass of the solid region Q of the density ρ bounded by the graphs of the equations ρ(x,y,z)=k, Q:2x+3y+6z=12,x=0, y=0, z=0

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Chapter 14.6, Problem 36E

Chapter 14.6, Problem 38E

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Chapter 14 Solutions

Calculus: Early Transcendental Functions (MindTap Course List)

Ch. 14.1 - Evaluate the iterated integral: 0433cosrdrd Ch. 14.1 - Prob. 1E Ch. 14.1 - Evaluate the integral: xx2yxdy Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Evaluate the integral: 04x2x2ydy Ch. 14.1 - Evaluate the integral: x3x(x2+3y2)dy Ch. 14.1 - Evaluate the integral: eyyylnxxdx;y0 Ch. 14.1 - Evaluate the integral: 1y21y2(x2+y2)dx Ch. 14.1 - Evaluate the integral: 0x2yeyxdy

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The mass and indicated coordinate of the centre of the mass of the solid region Q of the density ρ bounded by the graphs of the equations ρ ( x , y , z ) = k , Q : 2 x + 3 y + 6 z = 12 , x = 0 , y = 0 , z = 0

Solution Summary: The author explains the formula used to calculate the total mass of the solid region and the moment of inertia.

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To Calculate: The mass and indicated coordinate of the centre of the mass of the solid region Q of the density ρ bounded by the graphs of the equations ρ(x,y,z)=k, Q:2x+3y+6z=12,x=0, y=0, z=0

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Chapter 14 Solutions