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xonycutieoxl1 2022-06-29 Answered
Triple integration x 2 + y 2 + z 2 2 z x 2 y 2 d x d y d z
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Answers (1)

Blaine Foster
Answered 2022-06-30 Author has 33 answers
It's better to use spherical coordinates
The region is x 2 + y 2 + z 2 2 z , i.e., x 2 + y 2 + ( z 1 ) 2 1
So we can define the spherical coordinates r , θ , ϕ such that
x = r sin θ cos ϕ
y = r sin θ sin ϕ
z = 1 + r cos θ
And the integral becomes
I = r = 0 1 θ = 0 π ϕ = 0 2 π r 6 sin 5 θ cos 2 ϕ sin 2 ϕ d ϕ d θ d r
Integrating with respect r and with respect to ϕ is straight forward, and the integral reduces to
I = π 28 θ = 0 π s i n 5 θ d θ
The integral with respect to θ is solved using the substitution u = cos θ, then
sin 5 θ d θ = ( 1 u 2 ) 2 d u = ( u 2 3 u 3 + u 5 5 )
Hence,
I = 4 π 105
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