Thumbnail: Comparison between 2-point Gaussian and trapezoidal quadrature. The blue line is the polynomial , whose integral in [−1, 1] is 2/3. The trapezoidal rule returns the integral of the orange dashed line. The 2-point Gaussian quadrature rule returns the integral of the black dashed curve. Such a result is exact, since the green region has the same area as the red regions. Image used with permission (CC BY-Sa 4.0; Paolostar).