# 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!. – Babak S. Aug 4 '13 at 8:14

You can also use the following limits indicating that we are using Cartesian coordinates. I use the symmetric of the solid volume as well.

$$4\int_{x=0}^{\sqrt{8}}\int_{z=0}^{\sqrt{8-x^2}}\int_{y=0}^2dydzdx$$

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Great (+1). You've plotted this shape in MAPLE too ? – Mahdi Khosravi Aug 3 '13 at 17:06
Is it available for free? – Mahdi Khosravi Aug 3 '13 at 17:08
@MahdiKhosravi: Of course not, but you can a preety workable one in Pasje Reza, chahrahe vali asr. It is fun working with it. Great in modeling, Mahdi. Jun mide bara intor kara. – Babak S. Aug 3 '13 at 17:11
I was looking for such a software for years! seeing your drawings made me really feel "It is possible to draw such a shapes!". I will purchase it ASAP. – Mahdi Khosravi Aug 3 '13 at 17:17
@BabakS.: Nice graphics again! +1 – Amzoti Aug 4 '13 at 1:17