# Parabolic cylindrical coordinates

Can someone, please, show me an example of when you are better of with parabolic cylindrical coordinates than with cartesian coordinates when computing a triple integral over a solid?

Things with symmetry around an axis, like a cylinder. Try this one with cartesian coordinates, then try it with cylindrical coordinates.

$$\int_{-2}^{2} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{\sqrt{x^2+y^2}}^{2} \,\,\,(x^2+y^2)\,\,dz\,dy\,dx$$

Last edited:
D H
Staff Emeritus
Things with symmetry around an axis, like a cylinder. Try this one with cartesian coordinates, then try it with cylindrical coordinates.

$$\int_{-2}^{2} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{\sqrt{x^2+y^2}}^{2} \,\,\,(x^2+y^2)\,\,dz\,dy\,dx$$

That works great in (circular) cylindrical coordinates, but not in parabolic cylindrical coordinates.

hehe... sorry man.
I must have read it too quickly. Honestly, I've never worked in parabolic cylindrical coordinates.

have a good one