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## Homework Statement

Ok so I solved the problem, I think. I would just like to check my work.

So the problem is:

Use polar coordinates to find the volume of the given solid bounded by the paraboloids z = 3x^2 + 3y^2 and z = 4 - x^2 - y^2.

## Homework Equations

r^2 = x^2 + y^2

x = r cos Θ

y = r sin Θ

## The Attempt at a Solution

Ok so I starts by setting the two equations equal to each other to find their intersection.

3x^2 + 3y^2 = 4 - x^2 - y^2

4x^2 + 4y^2 = 4 (add the x^2 and y^2 across)

x^2 + y^2 = 1 (divide by four)

So then that the paraboloids intersect in the circle x^2 + y^2 = 1.

So from this, I figured that the area I was integrating over in polar coordinates was 0 ≤ r ≤ 1 and 0 ≤ Θ ≤ 2pi.

Then, I converted the two equations to polar coordinates:

z = 3x^2 + 3y^2 = 3r^2

z = 4 - x^2 - y^2 = 4 - r^2

Then I took the double integral of (4 - r^2 - 3r^2) r dr dΘ from 0 to 1 (with respect to r) and then from 0 to 2pi (with respect to Θ).

I subtracted the two r^2 and mulitplied to other r through so that I was now taking the double integral of 4r - 4r^3 dr dΘ over the same area.

I integrated that to get 2r^2 - r^4 and plugged in 1 and 0 for the values of r and got 1.

Then I integrated 1 from 0 to 2pi to get a final answer of 2pi.

Did I do something wrong?

Thanks! :)