# Setting up a triple integral to find volume of a region

1. Nov 16, 2009

### yolanda

1. The problem statement, all variables and given/known data

I need to set up the triple integral to find the volume of the region bounded by the sphere: x^2+y^2+z^2=a^2 and the ellipsoid: (x^2/4a^2)+(4y^2/a^2)+(9z^2/a^2)=1.

2. Relevant equations

above

3. The attempt at a solution

I'm not sure which interval I should be using here. I made a 3D graph of the region, but the a variable is really throwing me off. Can anyone point me in the right direction here? Thanks in advance!

2. Nov 16, 2009

### tnutty

In what type of coordinates? Spherical coordinates looks good.

3. Nov 16, 2009

### yolanda

I was considering spherical as well. That would work. I'm not picky on which coordinate system we use.

4. Nov 16, 2009

To calculate the volume of the sphere which is x2 + y2 + z2 = a2 you should use the following for the cartesian coordinate.

-a $$\leq$$ z $$\leq$$ a

-$$\sqrt{a^2 - z^2}$$ $$\leq$$ y $$\leq$$ $$\sqrt{a^2 - z^2}$$

-$$\sqrt{a^2 - y^2 - z^2}$$ $$\leq$$ x $$\leq$$ $$\sqrt{a^2 - y^2 - z^2}$$

And you can do it the same for the ellipsoid!

5. Nov 17, 2009

### yolanda

So, for the ellipsoid, should it be the following?:

-a<=z<=a
-sqrt[-x^2-4(9*z^2-a^2)]/4 <=y<= sqrt[-x^2-4(9*z^2-a^2)]/4 (shouldn't have x values...hmmm)

I'm a bit lost here Would someone mind setting up the triple integral for me? I think having a look at the final product would open up some doors in my brain. No need to evaluate. Thanks again, people!

6. Nov 17, 2009

Vsphere = $$\int$$$$^{a}_{-a}$$$$\int$$$$^{\sqrt{a^2 - z^2}}_{-\sqrt{a^2 - z^2}}$$$$\int$$$$^{\sqrt{a^2 - y^2 - z^2}}_{-\sqrt{a^2 - y^2 - z^2}}$$dx dy dz

and for the ellipsoid $$\frac{-a}{3}$$ $$\leq$$ z $$\leq$$ $$\frac{a}{3}$$ and ...

7. Nov 17, 2009

### yolanda

Oh ok, I think I get that part now.

-a/3 <= z <= a/3

-sqrt[(a^2-9z^2)/4] <= y <= sqrt[(a^2-9z^2)/4]

-sqrt[4a^2-16y^2-36z^2] <= x <= sqrt[4a^2-16y^2-36z^2]

So, now that I have my intervals for both shapes, what comes next? I need the volume of the ellipsoid, but only the part that's within the sphere. Sorry to ask so many questions, but my book has no similar examples that I can work with. I really appreciate your help.

8. Nov 18, 2009

V= $$\int$$ $$\int$$ $$\int$$ r2sin$$\phi$$ dr d$$\theta$$ d$$\phi$$
0 $$\leq$$ r $$\leq$$ $$\frac{4 \sqrt{2}a}{sin\phi \sqrt{12 cos^2(\theta) +20}}$$
0 $$\leq$$ $$\theta$$ $$\leq$$ 2 $$\pi$$
0 $$\leq$$ $$\phi$$ $$\leq$$ $$\pi$$