Difference between Quadrature Rules and Riemann Sum I am reading and learning about Quadrature rules. Would it be correct to say that a Riemann Sum method enters in the category of quadrature rules? I am asking because the Riemann sum evaluates the "area under the curve by a series of small rectangles" and in the with quadrature rules you seem to be able to do do the same thing? I find strange the wikipedia page on numerical integration doesn't say more about the Riemann sum.
Thank you.

By Riemann sum methods, I suspect you mean estimating the area by a finite number of rectangles. This is a sort of quadrature, although it is perhaps the 'trivial' sort of quadrature. Good quadratures, such as Gaussian Quadrature have great error estimates because they involve clever weighting schemes. An $n$-term Gaussian quadrature has zero error on degree $2n-1$ polynomials, for example. But $n$-term Riemann sums have zero error on linear functions, and have error on everything else in general.
If you want to know more about using Riemann sums to approximate integrals, you should look into the rectangle method. Really, though, this is equivalent to the 'midpoint approximation' method that many people learn in their first semester of calculus. This is the first of the Newton-Cotes formulas (first meaning the worst). The second is Simpson's Rule - something else that many students learn in first semester calculus.
The wiki article on numerical integration mentions both the rectangle method and Newton-Cotes, and those pages are perfectly adequate. But it doesn't give them more credence because they are comparatively worse and harder to calculate.
