MHB Vector Calculus - Cylindrical Co-ordinates

[RigB]

I have a question, to use cylindrical coordinates to find the volume of the ellipsoid ${R}^{2}+{3z}^{2}=1$.

I know for cylindrical coordinates the Jacobian is $r$ so I have some integral:

$$\iiint (r)dzdrd\theta$$

However I am struggling to work out the bounds of the integral for $z,r,\theta$ and also what I am integrating. Please may someone explain the method for working out the limits in this example and what I am integrating? I should be okay to do the integral itself from then on.

I would really appreciate it. Thank you.

 

[Fantini]

I think there's some information missing here. What you have so far is an ellipse in $R$ and $z$.