# Polar Coordinates and volume?

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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.

• 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

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