Trapezium rule

The trapezium rule (or trapezoid rule) is a method used for approximating definite integrals. Formally speaking, it’s not a type of Riemann sum because it uses a different principle.

$$T_n=\frac{1}{2}\Delta x(f(x_0)+2f(x_1)+\cdots +2f(x_{n-1})+f(x_n))$$ $$T_n=\frac{1}{2}(L_n+R_n)$$ $$\underset{n\to \infty }{\lim }{T}_n={\int }_a^{b}f(x)dx$$

We observe that the trapezium rule can approximate definite integrals far better than left or right end-point Riemann sums. Generally speaking it tends to be less accurate than the midpoint rule.