[hgsgm]

Huh? Quadrature is a general term for "measuring area". In this context it's a synonym for integration.

I think you are trying to say that it's better to do weighted sums of fewer samples, instead of uniformly weighted Reimann sum. Both are "calculus definition" integration, of course, since calculus is true.

[ouid]

When mathematicians say Quadrature, they mean that if your function is suitably approximated by projecting onto some orthogonal basis functions, you can get very cheap approximations by cleverly expressing those integrals exactly as a linear combination of their values at certain points along the interval. You need very few.

https://en.m.wikipedia.org/wiki/Gaussian_quadrature

It is significantly more subtle than what you are thinking.

[hgsgm]

From that page:

> In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration

Simpson's rule, taught in first year calculus, is exact for cubics, with 3 sample points.