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.
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?