# Find volume of region using change of variables

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?

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## 1 Answer

Your region is as follows:

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

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er why isn't r from 0.5 to 1? (plugging in z=1 and z = 4 into the equations) –  praks5432 Nov 1 '12 at 6:33
and what about t? what's t? –  praks5432 Nov 1 '12 at 6:35
@praks5432: Which coordinates are you using? Cartesian $(x,y,z)$? Cylindrical $(r,\theta,z)$? Or Spherical $(\rho,\theta,\phi)$? It looks to me that you are using the second coordinates, so we have 3 coordinates $(r,\theta,z)$ not 4 coordinates. I mean that existing another parameter, say $t$, is meaningless here. –  B.S. Nov 1 '12 at 6:44
Beautiful! $\quad +1 \;\ddot\smile\;$ I hope to see you tonight before I go to bed (my time that is, but morning your time!) –  amWhy Apr 2 at 2:48
@amWhy: I came here but seems you went to the Heaven. Have a peaceful colored sleep my dear Amy. –  B.S. Apr 2 at 6:26
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