(a) Explanation Given: The solid is the hemisphere of radius a. Formula used: If f is a spherical region E given by a ≤ ρ ≤ b , α ≤ θ ≤ β , c ≤ ϕ ≤ d , then, ∭ E f ( x , y , z ) d V = ∫ α β ∫ a b ∫ c d f ( ρ sin ϕ cos θ , ρ sin ϕ sin θ , ρ cos ϕ ) ρ 2 sin ϕ d ϕ d ρ d θ (1) If g ( x ) is the function of x and h ( y ) is the function of y and k ( z ) is the function of z then, ∫ a b ∫ c d ∫ e f g ( x ) h ( y ) k ( z ) d z d y d x = ∫ a b g ( x ) d x ∫ c d h ( y ) d y ∫ e f k ( z ) d z (2) The spherical coordinates ( ρ , θ , ϕ ) corresponding to the rectangular coordinates ( x , y , z ) is, ρ = x 2 + y 2 + z 2 ϕ = cos − 1 ( z ρ ) θ = cos − 1 ( x ρ sin ϕ ) Calculation: From the given condition, it is observed that ρ varies from 0 to a , θ varies from 0 to 2 π and ϕ varies from 0 to π 2 . Then, by the equation (1), the mass of the solid is, m = ∫ 0 2 π ∫ 0 π 2 ∫ 0 a K d z d y d x = ∫ 0 2 π ∫ 0 π 2 ∫ 0 a K ( ρ 2 sin ϕ ) d ρ d ϕ d θ = K ∫ 0 2 π ∫ 0 π 2 ∫ 0 a ( ρ 2 sin ϕ ) d ρ d ϕ d θ Use the equation (2) to separate the integrals and integrate it. m = K ∫ 0 2 π d θ ∫ 0 π 2 sin ϕ d ϕ ∫ 0 a ρ 2 d ρ = K [ θ ] 0 2 π [ − cos ϕ ] 0 π 2 [ ρ 3 3 ] 0 a = K [ 2 π − 0 ] [ − cos π 2 + cos ( 0 ) ] [ a 3 3 − 0 3 3 ] = 2 K π ( − 0 + 1 ) ( a 3 3 ) = 2 3 K π a 3 Thus, the mass of the given solid is 2 3 K π a 3 (b) To determine To find: The moment of inertia about the diameter of its base for the region in part (a).

Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112280

Author: James Stewart

Publisher: Cengage Learning

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Chapter 12.7, Problem 31E

(a)

To determine

To find: The centroid of the solid in example 4 by using spherical coordinates.

(b)

To determine

To find: The moment of inertia about the diameter of its base for the region in part (a).

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Chapter 12.7, Problem 30E

Chapter 12.7, Problem 32E

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Chapter 12 Solutions

Essential Calculus: Early Transcendentals

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The centroid of the solid in example 4 by using spherical coordinates.

Solution Summary: The author explains how to find the centroid of the solid by using spherical coordinates.

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To find: The centroid of the solid in example 4 by using spherical coordinates.

To find: The moment of inertia about the diameter of its base for the region in part (a).

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Chapter 12 Solutions