Question

Use spherical coordinates. Evaluate(x2 + y2) dV E where E lies between the spheres x2 + y2 + z2 = 1 and x2 + y2 + z2 = 25.

Answer 1

J = ∫∫∫[E] z dV 

Substitute with spherical coordinates: x=r*cosθ*sinφ, y=r*sinθ*sinφ, z=r*cosφ, x²+y²+z²=r², dV=r²sinφ drdθdφ, 
the equation of the sphere x²+y²+z²=4 turns to r=2, 
the equation of the sphere x²+y²+z²=16 turns to r=4, 
so the volume of integration in spherical coordinates will be limited by 
0<φ<π/2, 0<θ<π/2, 2<r<4 

J = ∫[0,π/2] ∫[0,π/2] ∫[2,4] r*cosφ r²sinφ drdθdφ = 
∫[0,π/2] dθ ∫[0,π/2] sinφ*cosφ dφ ∫[2,4] r³ dr = 
(π/4)*∫[0,π/2] sin(2φ) dφ ∫[2,4] r³ dr = 
(π/4)*[-cos(2φ)/2]*r⁴/4 [φ=0 to π/2, r=2 to 4] = 
(π/4)*(1/2 +1/2)*(4⁴/4 -2⁴/4) = (π/4)*(64 -4) = 15π

Answer 2

✒ Here is the Answer ✍✍

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⭐⭐the answere is 15π ⭐⭐