# Find the volume of a cylinder laying on ZX plane

I need to find the volume of an object restricted with the $x^{2}+z^{2} < 8$ and $0 < y < 2$ planes. It would be easy if the cylinder were "parallel" to the XY plane, because then:

$$0 < r < 2\sqrt{2}$$ $$0 < \phi < 2\pi$$ $$0 < z < 2$$

But well, how should I handle this here?

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rename your variables, or change the definition of $r,\phi$ ! –  Maesumi Aug 3 '13 at 14:03

Think of this as a cylinder whose base lies in the $x$-$z$ plane, encompassed by the region $x^{2}+z^{2} < r^2 = 8$ and $0 < y < 2$, so that:

$$0 < r < 2\sqrt{2}$$ $$0 < \phi < 2\pi$$ $$0 < y < 2$$

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Even following the OP's line of thought/approach +1 –  Amzoti Aug 4 '13 at 1:16
Thanks, @Amzoti! –  amWhy Aug 4 '13 at 1:17
@amWhy: I love extended polar coordinates than historical Cartesian ones, so it needs another TU!. –  B. S. Aug 4 '13 at 8:14
$$4\int_{x=0}^{\sqrt{8}}\int_{z=0}^{\sqrt{8-x^2}}\int_{y=0}^2dydzdx$$