As for the demo, it does give the correct answer, and also for the other cases I checked. Clean up the documentation (as John eluded to) and don't have the demo "clear all", then I may give a higher rating. It does integrate what is considered a 'general region' for this type of problem. Note also that two of the edges need not be parallel, as in integrating f(x,y) over the domain x^3<y<sqrt(x) for 0<x<1.
I was looking forward to a useful tool, but was left highly disappointed. The author has just copied dblquad, renaming it dblquadx. Apparently the arguments now mean different things, but the help is the old help, which is not only meaningless in this function, but now totally incorrect. At least the MathWorks copyright notice was also copied over.
I'll also comment that even with correct help, the website description is still incorrect, as the domain is not general at all. It has a very specific pseudo-trapezoidal shape, where two of the edges of the polygon are parallel, and the other two edges are general curvilinear shapes. I have not verified that it works or not.