Setting up a triple integral to find volume of a region

[yolanda]

Homework Statement

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.

Homework Equations

above

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!

 

[tnutty]

In what type of coordinates? Spherical coordinates looks good.

 

[yolanda]

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

 

[rado5]

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!

 

[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!

 

[rado5]

V_sphere = \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 ...

 

[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.

 

[rado5]

I solved it in spherical coordination and I think it is correct, if it is not, somebody please tell me why?

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