(a) Explanation 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 example 4, it is known that m = π 8 K . From the symmetry of the given region, it is observed that the M y z and M x z are equal to 0. The value of M x y is given by, M x y = ∫ 0 2 π ∫ 0 π 4 ∫ 0 cos ϕ K z d z d y d x = K ∫ 0 2 π ∫ 0 π 4 ∫ 0 cos ϕ ρ cos ϕ ( ρ 2 sin ϕ ) d ρ d ϕ d θ = K ∫ 0 2 π ∫ 0 π 4 ∫ 0 cos ϕ ρ 3 cos ϕ sin ϕ d ρ d ϕ d θ Integrate with respect to ρ and apply the limit of it (b) To determine The moment of inertia about the z axis.

8th Edition

ISBN: 9781305266643

Author: James Stewart

Publisher: Cengage Learning

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Chapter 15.8, Problem 31E

(a)

To determine

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

(b)

To determine

The moment of inertia about the z axis.

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Chapter 15.8, Problem 30E

Chapter 15.8, Problem 32E

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Chapter 15 Solutions

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

Solution Summary: The author analyzes the centroid of the solid in example 4 by using spherical coordinates.

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

The moment of inertia about the z axis.

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Chapter 15 Solutions