a. Explanation Given information: The hemisphere z = 9 − x 2 − y 2 whose the density function is ρ ( x , y , z ) = x y + z . Formula used: The coordinates of centroid are denoted by ( x ¯ , y ¯ , z ¯ ) . Since the density function is constant and solids are symmetrical, that is x ¯ = y ¯ = 0 and the z ¯ is given by the formula, z ¯ = M x y m = ∭ D z r ρ d z d r d θ ∭ D r ρ d z d r d θ Where m is the mass of the solid lying in the region and ρ is a constant. The coordinates in rectangular form are denoted by ( x , y , z ) and coordinates in cylindrical form are denoted by ( r , θ , z ) . To convert the rectangular coordinates to cylindrical coordinates use the following conversions: r = x 2 + y 2 tan θ = y x z = z To convert the cylindrical coordinates to rectangular coordinates use the following conversions: x = r cos θ y = r sin θ z = z Calculation: The coordinates of centroid are denoted by ( x ¯ , y ¯ , z ¯ ) . Since the density function is constant and solids are symmetrical, that is x ¯ = y ¯ = 0 and the z ¯ is given by the formula, z ¯ = M x y m = ∭ D z r ρ d z d r d θ ∭ D r ρ d z d r d θ Where m is the mass of the solid lying in the region and ρ is a constant. Therefore, the mass of the hemisphere is given by the integral ∭ D r ρ d z d r d θ . For the hemisphere z = 9 − x 2 − y 2 whose the density function is ρ ( x , y , z ) = x y + z b. To determine To calculate: The integration formula for x -coordinate of center of mass of the hemisphere z = 9 − x 2 − y 2 whose the density function is ρ ( x , y , z ) = x y + z . c. To determine To calculate: The integration formula for moment of inertia about z -axis of the hemisphere z = 9 − x 2 − y 2 whose the density function is ρ ( x , y , z ) = x y + z .

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 12.7, Problem 43PS

To determine

To calculate: The integration formula for mass of the hemisphere z=9−x2−y2 whose the density function is ρ(x,y,z)=xy+z .

To determine

To calculate: The integration formula for x -coordinate of center of mass of the hemisphere z=9−x2−y2 whose the density function is ρ(x,y,z)=xy+z .

To determine

To calculate: The integration formula for moment of inertia about z -axis of the hemisphere z=9−x2−y2 whose the density function is ρ(x,y,z)=xy+z .

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Chapter 12.7, Problem 42PS

Chapter 12.7, Problem 44PS

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Chapter 12 Solutions

Student's Solution and Survival Manual for Calculus

Ch. 12.1 - Prob. 1PS Ch. 12.1 - Prob. 2PS Ch. 12.1 - Prob. 3PS Ch. 12.1 - Prob. 4PS Ch. 12.1 - Prob. 5PS Ch. 12.1 - Prob. 6PS Ch. 12.1 - Prob. 7PS Ch. 12.1 - Prob. 8PS Ch. 12.1 - Prob. 9PS Ch. 12.1 - Prob. 10PS

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The integration formula for mass of the hemisphere z = 9 − x 2 − y 2 whose the density function is ρ ( x , y , z ) = x y + z .

Solution Summary: The author explains the integration formula for mass of the hemisphere z=sqrt9-x2- y2. Since the density function is constant and solids are

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To calculate: The integration formula for mass of the hemisphere z=9−x2−y2 whose the density function is ρ(x,y,z)=xy+z .

To calculate: The integration formula for x -coordinate of center of mass of the hemisphere z=9−x2−y2 whose the density function is ρ(x,y,z)=xy+z .

To calculate: The integration formula for moment of inertia about z -axis of the hemisphere z=9−x2−y2 whose the density function is ρ(x,y,z)=xy+z .

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Chapter 12 Solutions