# Polar Double Integrals

1. Dec 4, 2013

### Lamebert

1. The problem statement, all variables and given/known data

The plane z = 2 and the paraboloid z = 8 − 6x2 − 6y2 enclose a solid. Use polar coordinates to find the volume of this solid.

2. Relevant equations

∫∫R f(x,y) dA = ∫βαba f(rcosθ, rsinθ) r dr dθ

3. The attempt at a solution

z = 2, z = 8 − 6x2 − 6y2

Setting these two equal, we can find where the two functions intersect.

2 = 8 − 6x2 − 6y2

so

0 = 6 − 6x2 − 6y2

Solving for x and y, we get

1 = x2 + y2

So the intersection is a circle of radius 1 on the plane z = 2.

Knowing this, we can write the domain of x and y both in terms of r and θ:

{ r,θ | 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2∏}

Using this domain, I set up my double integral with the above layout found in my provided equations section. I don't feel like typing all the integrations out, but is my above process wrong? If not, I can focus on finding errors in my integration and ask further questions as needed.

Thanks.

Last edited: Dec 4, 2013
2. Dec 5, 2013

### haruspex

Looks to me that you are in danger of including the volume below z=2 (within the circle).