Gauss-Legendre vs trapezoid quadrature

What you are seeing: two ways of approximating $\int_{-1}^{1} f(x)\, dx$. The trapezoidal rule uses $n + 1$ equispaced points and converges as $O(h^2)$ for smooth $f$. Gauss-Legendre uses $n$ optimized nodes (roots of $P_n$) with matching weights, exact for polynomials up to degree $2n - 1$. For analytic $f$ the GL error decays exponentially.

Top panel: $f(x)$ with nodes overlaid (orange = trapezoid, cyan = GL). Bottom: log-error vs $n$. For smooth $\cos(2x)$, GL reaches machine precision by $n = 16$ while trapezoid still loses accuracy as $O(h^2)$.