Rather than adding the integrals, I think you actually need to use an iterated integral since you're concerned with both the x and y directions simultaneously. In this case, it will obviously be a double integral. When setting it up, you should make y the inner integral since the general horizontal line across the region goes between 2 functions. The x part makes a good candidate for the outer integral because you can then just integrate from 0 to the constant b.
This page has a number of examples involving triangles.