Double integral in polar coordinates

[simmonj7]

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! :)

 

[Gregg]

I don't think you have done this properly. I think that you need to find the volume of the solid for a surface which is

$$z=4(-x^2-y^2+1)$$

for z>0

Which intersects the xy-plane on the unit circle centred at (0,0)

 

[simmonj7]

That equation you gave is not in the problem I stated so I have no idea where you are coming from.

 

[Gregg]

Well you said that you need to find the volume between the two surfaces?

$$z_1 = 3x^2 + 3y^2$$
$$z_2 = -x^2 - y^2 + 4$$

The difference in their height at every point is

$$z=z_2-z_1=-4x^2-4x^2+4$$

I think that the solution will be to find the area under the surface above which can be written as

$$z=4(-x^2-y^2+1)$$

that is, the volume between this curve and the xy-plane. Also, to find a volume, polar co-ordinates are not sufficient, how can you get a volume when integrating for a 2D problem? Use cyclinderical polar coordinates or spherical polar coordinates.

Also, I suggest the former method, adding small disks of varying radius and infinitesimal height dz.

 

[simmonj7]

I can only use polar coordinates cause that is the unit we are doing in the book so they have to be sufficient.

 

[simmonj7]

And you were saying that I was trying to find the area under the surface z = 4(-x^2 - y^2 +1).
However, if you look, I did that because I integrated 4 - 4r^2 which is exactly that.

 

[Gregg]

Then I think actually it should be OK

 

[LCKurtz]

Did I do something wrong?
Thanks! :)

No. It is exactly right.