Choosing which technique to use to solve multiple integrals? There are at least two ways to solve double integrals. One way is to use interated integrals, based on Fubini's, or similar theorems. The other way is to reduce the double integral to a line (curve) integral using something like Green's theorem.
How do you choose which technique to use? Green's theorem appears to work best in a conservative force field, that is path independent. So do you identify such situations by seeing whether the associated partial derivatives integrate into exact equations?  Or does it have something to do with limits of integration?
Ditto for choosing between iterated triple integrals and surface integrals using e.g. Stokes' Theorem.
 A: It pretty much just depends on the problem! I like to stick with line integrals in the plane, however its not always best especially when the area is expressed easily as double integral.  Same reasoning applies to Stoke's theorem and the Divergence theorem.  However, I almost always prefer to work with volume integrals over surface integrals with respect to the divergence theorem.  In the end I always end up doing a bit of analysis on the problem, i.e. examining the vector field and surface before making a choice.
A: When you are talking about things like Green's theorem, Stokes' Theorem, etc., you are talking about a flow of matter from a bounded area or volume across a curve or surface represented by a membrane.
There are two ways to measure this "flow." One is to measure the changes in matter within each unit of volume or area and integrate them. The other is to measure the amount of "flow" across the boundary.
Because it is one order of magnitude lower, measuring and integrating "flow" across the curve, boundary, surface or membrane is usually easier than measuring and integrating flow from each area or volume, all other things being equal.
The "all other things being equal" fails to hold when there are "singularities," or leaks, in the area or surface, often involving the origin (the zero vector), or some point that maps to the origin. That's because matter is not "conserved" (within the relevant region), so measuring flow across boundaries doesn't work. In these cases, the higher order integral works better as a measure of flow, even if it is the more difficult computation.
