Double Integrals: Finding the Volume of a Solid Using Polar Coordinates

[Lamebert]

Homework Statement

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

Homework Equations

∫∫_R f(x,y) dA = ∫^β_α∫^b_a f(rcosθ, rsinθ) r dr dθ

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.

 

[haruspex]

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