Find the Volume .

Question

Find the volume of $(x/3)^2 + (y/3)^2 + (z/2)^2 = 36$ bounded by $x^2 + y^2 - 2y = 0$ and the plane $z=0$. was an exercise on my exams and im not sure i got it right. Sorry if im making you do the calculations but i wanted to see if there is an easy way to calculate it or it has lots and lots of arithmetic operations .

Answer 1

Hint let $x=18\rho\cos\theta\sin\phi$ , $y=18\rho\sin\theta\sin\phi$ and $z=12\rho\cos\phi$ we have $$dz\,dx\,dy=3888\,\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$$ and $${{\left( \frac{x}{3} \right)}^{2}}+{{\left( \frac{y}{3} \right)}^{2}}+{{\left( \frac{z}{2} \right)}^{2}}=36\Leftrightarrow \,{{\rho }^{2}}=1$$