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## 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 [tex]\frac{x^2}{4a^2}[/tex] + [tex]\frac{4y^2}{a^2}[/tex] + [tex]\frac{9z^2}{a^2}[/tex] = 1

## Homework Equations

## The Attempt at a Solution

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

V= [tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex] r

^{2}sin[tex]\phi[/tex] dr d[tex]\theta[/tex] d[tex]\phi[/tex]

0 [tex]\leq[/tex] r [tex]\leq[/tex] [tex]\frac{4 \sqrt{2}a}{sin\phi \sqrt{12 cos^2(\theta) +20}}[/tex]

0 [tex]\leq[/tex] [tex]\theta[/tex] [tex]\leq[/tex] 2 [tex]\pi[/tex]

0 [tex]\leq[/tex] [tex]\phi[/tex] [tex]\leq[/tex] [tex]\pi[/tex]

How I found r? Well we have two z

^{2}s here, one for the ellipsoid and the other for the sphere. I actually found r by equating the tow z

^{2}s! And so I got 0 [tex]\leq[/tex] r [tex]\leq[/tex] [tex]\frac{4 \sqrt{2}a}{sin\phi \sqrt{12 cos^2(\theta) +20}}[/tex]