Why Does Gaussian Integration Work?

[Faceless Master]

Hi all,
why does Gaussian Integration in one dimension with n points integrate exactly with a polynomial of order 2n-1 ?

thanks

 

[HallsofIvy]

Because Gaussian integration, on n points, effectively approximates the integrand by a polynomial function passing through those n points. And that polynomial is a polynomial of degree n-1.

If the integrand itself is a polynomial of degree n-1, the "approximation" Gaussian integration uses would be the integrand and so would give an exact result.