Let
be the tetrahedral region
{ }
Evaluate
Well, if we know that x, y and z are all greater then zero and that they are also bounded by a function, in this case that tells us something aboute the region.
Captain Black has decided to evaluate z first, y next and x last so we will keep that order for consistency (it actually doesn't matter).
Notice if we re-arrange the equation we can get,
That physically means that the function is our upper bound for z! And since we know that our lower bound is 0, we can say
Certainly we can do the same for y realizing that in the x-y domain z = 0,
And
Thus,
Our last variable, x, has to have constant bounds. Well, we know the lower bound is 0, but what about the upper bound? the upper bound happens when both y and z are 0, which from the above equation would yield 1. Thus,
All together this yields what Captain black had already posted. On how to evaluate this you evaluate the first integral (dz integral ) then the middle integral (dy integral) then the last integral (dx integral).