Use cylindrical or spherical coordinates, whichever seems more appropriate.?

Question

Evaluate ∫∫∫ z dV which lies above the paraboloid z = x^2 + y^2 and below the plane z=6y. Use a computer algebra system to evaluate the integral.

I don't know how to use a computer algebra system to evaluate the integral. Can someone help me with that?

Answer 1

Favorite Answer

Use cylindrical coordinates:

We have z = r^2 and z = 6r sin θ.

These intersect with r = 6 sin θ, a circle completely traced out with θ in [0, π].

So, the volume ∫∫∫ 1 dV equals

∫(θ = 0 to π) ∫(r = 0 to 6 sin θ) ∫(z = r^2 to 6r sin θ) 1 * (r dz dr dθ)

(This can be entered in a computer algebra system as needed.)

Evaluating this directly:

∫(θ = 0 to π) ∫(r = 0 to 6 sin θ) r(6r sin θ - r^2) dr dθ

= ∫(θ = 0 to π) ∫(r = 0 to 6 sin θ) (6r^2 sin θ - r^3) dr dθ

= ∫(θ = 0 to π) (2r^3 sin θ - r^4/4) {for r = 0 to 6 sin θ} dθ

= ∫(θ = 0 to π) 108 sin^4(θ) dθ

= ∫(θ = 0 to π) 27 (1 - cos(2θ))^2 dθ, half angle identity

= ∫(θ = 0 to π) 27 (1 - 2 cos(2θ) + cos^2(2θ)) dθ

= ∫(θ = 0 to π) 27 (1 - 2 cos(2θ) + (1/2) (1 + cos(4θ))) dθ

= ∫(θ = 0 to π) 27 (3/2 - 2 cos(2θ) + (1/2) cos(4θ)) dθ

= 81π/2.

I hope this helps!