Q:

Thumb rules for changing order of integration

Are there any thumb rules to quickly find new limits of a double integral when changing the order of integration? I can get the answer by drawing the area but frankly, that has bit of a beginner's feel to it. I have occasionally seen people using Heaviside function too but not sure that speeds up things a lot. Happy to be corrected though.
So the question is how do the pros change the order of integration? Are there any thumb rules that they apply to quickly get the new limits? Even if only for some simplified cases that is still fine. Appreciate any help or pointers!

A:

Drawing a picture is the best way.  I had a colleague long ago who claimed that he taught his Calc students to change the order of integration (in both double and triple integrals) without drawing.  I investigated, and his technique turned out to be:  Choose very carefully designed problems.
So it's possible, in some special cases, to extract equations from the limits of integration and then solve for a different variable, and then have new, correct limits.  But hardly ever does this work.  
The single most important thing that caused the scientific part of the Renaissance was the invention of Cartesian coordinates.  By being able to translate between the two branches of math (Geometry and Algebra) we were able to solve $10^n$ times as many problems than ever before.  We call it "Analytic Geometry".  The whole point of Analytic Geometry is that when things get tough in Geometry, you can switch to Analysis.  And when things get tough in Analysis, you can switch to Geometry.  What I'm saying is:  Don't be afraid or embarrassed to draw the picture.  That's how we all do it.  
Also:  When you switch to polar coordinates, you also need to draw the picture.

