Explanation Given information: The solid bounded by the surface z = x 2 + y 2 and by the plane with equation z = 9 . 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 solid is bounded by the surface z = x 2 + y 2 , this is in rectangular coordinates ( x , y , z ) , to convert it in cylindrical coordinates ( r , θ , z ) , use the relation r = x 2 + y 2 z = x 2 + y 2 z = r 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. Since, the region D is a solid bounded by the surface z = x 2 + y 2 and by the plane with equation z = 9

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 41PS

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To calculate: The centroid of the solid bounded by the surface z=x2+y2 and by the plane with equation z=9 .

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Chapter 12.7, Problem 40PS

Chapter 12.7, Problem 42PS

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Chapter 12 Solutions

Student's Solution and Survival Manual for Calculus

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The centroid of the solid bounded by the surface z = x 2 + y 2 and by the plane with equation z = 9 .

Solution Summary: The author calculates the centroid of the solid bounded by the surface and the plane with equation z=9. Since the density function is constant and solids are symmetrical, the formula gives the coordinates in

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To calculate: The centroid of the solid bounded by the surface z=x2+y2 and by the plane with equation z=9 .

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Chapter 12 Solutions