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Integration with Cylindrical and Polar Coordinates

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For the transformation into cylindrical coordinates

$\displaystyle x = \varrho\cos\varphi,\quad y = \varrho\sin\varphi,\quad z = z $

the volume element has the form

$\displaystyle dx\,dy\,dz = \varrho d\varrho\,d\varphi\,dz \,. $

In particular the integral of a function $ f$ on a cylinder is given by

$\displaystyle Z: \quad 0 \le \varrho \le \varrho_0 \,, \quad 0 \le z \le z_0 $

$\displaystyle \int\limits_Z f = \int\limits_0^{z_0} \int\limits_0^{2 \pi} \int\limits_0^{\varrho_0} f( \varrho,\varphi,z) \ \varrho \ d\varrho\,d\varphi\,dz \,. $

Similarly the volume element for the transformation with respect to polar coordinates

$\displaystyle x = r \cos\varphi, \quad y = r \sin\varphi, \quad $

is

$\displaystyle dx\,dy = r dr\,d\varphi \,. $

()

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  automatically generated 8/ 4/2008