Regla del trapecio

Interpolando dos puntos $(x_{0}, f(x_{0}))$ y $(x_{1}, f(x_{1}))$, mediante un polinomio de Lagrange

\begin{equation*} f(x) = \begin{bmatrix} \cfrac{x - x_{1}}{x_{0} - x_{0}} \\ \cfrac{x - x_{0}}{x_{1} - x_{0}} \end{bmatrix}^{T} \begin{bmatrix} f(x_{0}) \\ f(x_{1}) \end{bmatrix} \end{equation*}

Integrando

\begin{equation*} \int_{x_{0}}^{x_{1}} f(x) \ dx = \begin{bmatrix} -\cfrac{1}{2} x_{0} + \cfrac{1}{2} x_{1} \\ -\cfrac{1}{2} x_{0} + \cfrac{1}{2} x_{1} \end{bmatrix}^{T} \begin{bmatrix} f(x_{0}) \\ f(x_{1}) \end{bmatrix} \end{equation*}