In one of my math courses we've just finished covering double integrals and our professor had mentioned that when evaluating double integrals, the work of evaluating certain double integrals will greatly simplify if you choose to integrate in "lumps" as oppose to "bits and pieces".
I'll illustate what I mean by lumps and pieces.
If we are given the following double integral,
This is an integral which would be much easier if we integrated it in lumps giving us,
However it's not always possible to integrate in lumps. For example if we had,
We could not preform an integration of the lump in this case, we must expand until our integral is in mutliple pieces and integrate the pieces individually.
Now what I'm trying to think of is how I can consistently identify and determine whether an given integral can integrated in a lump, or whether it must be done in pieces.
Is there a simple way of doing this? What are the trouble cases that we cannot integrate in a lump?