Use the cylindrical coordinates to find the volume

[hbomb]

I'm having trouble figuring out this volume.

Use the cylindrical coordinates to find the volume of the solid S bounded by z=x^2 + y^2 and
z=12 - 2x^2 - 2y^2

I've included a pic of what I think the regions and the solid look like

http://img.photobucket.com/albums/v87/hbombblack/3dgraphcopy.jpg

I think this is what the graph looks like. I would appreciate it if some one could show the steps of getting the volume, cause I'm kinda confused. I have an idea of what needs to be done, I just want to see if I'm on the right track before I get started.

 

[Hippo]

The integral you need to use will be of the form

$$\int \int \int r \ dz \ dr \ d \theta$$

Choosing your upper and lower limits should not be too difficult.

Here's a hint: which function z(r, theta) is above the other?

 

[hbomb]

I know all about cylindrical coordinates and how I should integrate. My question is when I integrate r. There's a constant radius where the two functions meet. So my question is, do I make two separate integrals? One for the top of the volume in question and another for the bottom. I solved the two system of equations and I came up with x^2 + y^2=4, which means that the radius is 2 where these two functions intersect. So for the first volume for dr I would do from x^2 + y^2 to 2. And for the second volume for dr I would do from 2 to 12 - 2x^2 - 2y^2. Is this correct? Or can I just go from x^2 + y^2 to 12 - 2x^2 - 2y^2?

 

[Hippo]

I'm having a hard time understanding your problem, sorry.

However, I think you'll find it simpler to disregard x and y completely. That way, you can deal with z as a function of r and theta.

The first such function is

$$z=r^2$$

the second

$$z=12-2{r^2}$$

Envision the r, z plane. In that plane, those two functions should intersect at (2, 4).

In the integral I showed you, the first integration is over z, not r. Integrate from the lower z to the upper z.

After that, integrate over r to receive the area in the r, z plane, which can be further integrated over theta.

Am I making any sense?

 

[hbomb]

Yes, it does make sense. What should my limits be for r though? That's the problem I'm having. The radius changes except for where the two functions intersect.

 

[Hippo]

Integrate r from 0 to 2.

It's a measure of distance from the z axis, so the lowest value it assumes is 0. The highest, in this case, is the point where the surfaces closes together, 2.

 

[Oggy]

It's
$$V=\int_0^{2\pi}d\phi \int_0^{2}rdr \int_{r^{2}}^{12-2r^{2}}dz=...=24\pi$$
Because the z=x^2+y^2 and z=12-2x^2-2y^2 intersect at r=2, and r increases from 0 to the maximum of 2.