six orders of integration

The $3!=6$ ways to order $dx$ , $dy$ , $dz$ in a triple integral — all give the same value, but differ in algebraic difficulty.

A triple integral has three variables to order, and $3!=6$ ways to arrange them: $dz\,dy\,dx$ , $dz\,dx\,dy$ , $dy\,dz\,dx$ , $dy\,dx\,dz$ , $dx\,dz\,dy$ , $dx\,dy\,dz$ . Every one computes the exact same number, since they're all different bookkeeping for the same solid's volume or weighted sum — but the algebra required can range from routine to nearly impossible depending on which order is chosen. A good order projects the solid onto the plane of the two outer variables to get a simple shadow, and checks the surfaces bounding the innermost variable are single-valued.

Examples

The tetrahedron bounded by the coordinate planes and $x+y+z=1$ has volume $\tfrac16$ whether set up as $\int_0^1\!\int_0^{1-x}\!\int_0^{1-x-y}dz\,dy\,dx$ or $\int_0^1\!\int_0^{1-z}\!\int_0^{1-y-z}dx\,dy\,dz$ .