I'm familiar with the integral. But why is that sum a particularly good approximation to the integral? It seems better than one has a right to expect, although it's been a while since I did any numerical analysis.
Generally I'd expect a numerical integration scheme to have an error like h^k for some constant k - for example for Riemann sums the error goes like h^2 and for Simpson's rule like h^5. h is the distance between the sampling points and so is analogous to 1/c. So this doesn't just follow from Riemann summation.
Other than that, it's just a Riemann sum presented in an aesthetically pleasing manner.
But the point is not to drop small terms at the extremes - it is still an infinite sum. The point is to approximate sections of the function by rectangles, and those rectangles are a bad approximation around 0 too.