I was trying to solve this question and I am pretty sure my result is correct, yet I was given wrong marks for it and the teacher refuses to elaborate. So I come to you:
Calculate the volume of an object created by intersection of a sphere
$$x^2+y^2+z^2\le5$$
and a cylinder
$$x^2+y^2\le3$$
Here is my approach: I am going to use cylindrical coordinates for this.
I know that the radius will be from $0$ to $\sqrt{3}$. Therefore
$$r \in [0,\sqrt{3}]$$
I also know that $\phi$ ranges from $0$ to $2\pi$.
$$\phi \in [0,2\pi] $$
And I know that $z$ ranges from the lower part of the sphere to the upper part of the sphere.
$$z \in [-\sqrt{5-x^2-y^2},\sqrt{5-x^2-y^2}]$$
Knowing that $x = r\cos{\phi}$ and $y = r\sin{\phi}$ and jacobian equal to $r$, I set up my integral.
$$ 2\int_{0}^{2\pi} \int_{0}^{\sqrt{3}} \int_{0}^{\sqrt{5-r^2}} r \,dz\,dr\,d\phi$$
The result was approximately $34.9844$ but I was told this is wrong. I couldn't find any online tools and this is driving me nuts. I will appreciate any help. Thank you.