When mathematicians say Quadrature, they mean that if your function is suitably approximated by projecting onto some orthogonal basis functions, you can get ver

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 https://en.m.wikipedia.org/wiki/Gaussian_quadrature It is significantly more subtle than what you are thinking.