Riemann sums are a type of numerical integration but you'd need a lot of function evaluations (i.e. rectangles) to get a fairly accurate answer. However if you approximate the area under a curve, say with trapezoids (the trapezoid rule) or parts of parabola's (Simpson's rule), with the same number of function evaluations you can get a far more accurate answer or use few function evaluations to get an anwer accurate to a certain tolerence.