Use spherical coordinates. Evaluate triple integral (x^2+y^2)dV, where E l

usagirl007A

usagirl007A

Answered question

2021-10-07

Put spherical coordinates to use. Consider the triple integral (x2+y2)dV, where E lies between the spheres x2+y2+z2=4 and x2+y2+z2=9

Answer & Explanation

Nathanael Webber

Nathanael Webber

Skilled2021-10-08Added 117 answers

So,
x2+y2=(ρsinϕ)2+(ρsinϕsinθ)2 
=ρ2sin2ϕ(cos2θ+sin2θ) 
=ρ2sin2ϕ
0π02π23(ρ2sin2ϕ)p2sinϕdpdθdϕ 
0π02π23ρ4sin3ϕdpdθdϕ 
0π02π15ρ5sin3ϕdθdϕ 
0π21152πsin3ϕdϕ 
Rewrite sin2=1cos2 
422π50πsinϕ(1cos2ϕ)dϕ 
u=cosϕ, du=sinϕdϕ 
422π511(1u2)du 
422π5111u2du 
422π5(u13u3)11=1688π15 
Result: 1688π15

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