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.

You're right that if you just 'truncate' (use a partial sum rather than the infinite sum) then the Riemann Sum is a 'numerical Quadrature'. However, this is also how integration is defined many times so it isn't reiterated at that point. Those rules from the Wikipedia page are just a more sophisticated way to do the numerical integration. There are many many ways. See for example the Euler Maclaurin_formula and it's use for numerical integration. 
A method favoured by Nick Trefethen is to approximate the function with chebyshev polynomials (by doing a discrete cosine transform), and then doing standard Gaussian Quadrature The chebyshev polynomials have many nice properties other than being orthogonal. For example they don't suffer from Runge's phenomenon. Have a look Trefethen's expositions on these topics - it is all very readable.
tl;dr
If your function is just a sequence of boxes this is a great idea, otherwise there are better methods.
