Find volume of region using change of variables

Question

I want to find the volume of the region $R$ that lies between

$$z= x^2 + y^2, \quad z= 4(x^2 + y^2), \quad z = 1, \quad z = 4$$

Using the transformation \begin{align} x &= \frac{r}{t}\cos(\theta)\\ y &= \frac{r}{t}\sin(\theta)\\ z &= r^2 \end{align}

Now, I understand how to do this problem(finding the jacobian, plugging in the transformation, doing the triple integral), but what I don't understand is how to find the bounds for r,t and theta. Is there a general method on how to do this?

Answer 1

Your region is as follows:

$r\in[2,4]$, $\theta\in[0..2\pi]$, $z\in[2..4]$