Triple integral problem?
In a homework assignment, I am asked to evaluate the triple integral of the region E for the function f(x,y,z)=z. Where E is bounded by z=(x^2)+(y^2) and z=4. I got an answer of 32*pi. With my limits of integration being, in cylindrical coordinates, {0<theta<2*pi, 0<r<2, 0<z<4}. I'm just wondering if my limits of integration are correct.
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The z-bounds are a bit off.
They should be z = x^2 + y^2 = r^2 to z = 4 (you forgot about the paraboloid).
So, the integral ∫∫∫ z dV equals
∫(t = 0 to 2π) ∫(r = 0 to 2) ∫(z = r^2 to 4) z * (r dz dr dt)
= 2π ∫(r = 0 to 2) ∫(z = r^2 to 4) rz dz dr
= 2π ∫(r = 0 to 2) (1/2)rz^2 {for z = r^2 to 4} dr
= π ∫(r = 0 to 2) r(16 - r^4) dr
= π ∫(r = 0 to 2) (16r - r^5) dr
= π (8r^2 - r^6/6) {for r = 0 to 2}
= 64π/3.
I hope this helps!