Use polar coordinates to find the volume of the given solid.
Bounded by the paraboloid z = 7 + 2×2 + 2y2 and the plane z = 13 in the first octant.
2 Answers

Using polar coordinates,
z = 7 + 2x^2 + 2y^2 = 7 + 2r^2.
This intersects the plane when 7 + 2r^2 = 13
==> r^2 = 3
==> r = √3, a circle.
(We take θ in [0, π/2], since we want the region in the first octant, where in particular x, y > 0.)
So, the volume equals
∫∫ (7 + 2x^2 + 2y^2) dA
= ∫(θ = 0 to π/2) ∫(r = 0 to √3) (7 + 2r^2) * (r dr dθ), converting to polar coordinates
= ∫(θ = 0 to π/2) dθ * ∫(r = 0 to √3) (7r + 2r^3) dr
= (π/2) * (7r^2/2 + r^4/2) {for r = 0 to √3}
= 15π/2.
I hope this helps!

kb, I am following your work, except..
when we get to this sage…
∫∫ (7 + 2x^2 + 2y^2) dA
shoud it not be?….
∫∫ 13 – (7 + 2x^2 + 2y^2) dA
and converting to cylindrical coordinates.
∫∫ (6 – 2r^2)r dr dθ
= (π/2) * (6r^2/2 – r^4/2) {for r = 0 to √3}
= 9 pi / 4