Hello everyone,

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 bylumpsandpieces.

If we are given the following double integral,

$\displaystyle

\int^{1}_{0} \int^{x-1}_{0} (4x + y)^{3}dydx$

This is an integral which would be much easier if we integrated it in lumps giving us,

$\displaystyle \frac{1}{4}\int_{0}^{1} [(4x+y)^{4}]^{x-1}_{0}dx$

etc...

However it's not always possible to integrate inlumps. For example if we had,

$\displaystyle

\int^{1}_{0} \int^{x-1}_{0} (4x + y^{2})^{3}dydx$

We could not preform an integration of thelumpin this case, we must expand until our integral is in mutliplepiecesand integrate thepiecesindividually.

Now what I'm trying to think of is how I canconsistently identify and determinewhether an given integral can integrated in alump,or whether it must be done inpieces.

Is there a simple way of doing this? What are the trouble cases that we cannot integrate in alump?

Thanks again!