Regra do Trapézio

The trapezoidal rule is an approximation for the area under a curve that is derived from inscribing several trapezoids under a curve. The area of the $i$th trapezoid is $\frac12h(b_i+b_{i+1})=\frac12(x_{i+1}-x_i)(f(x_i)+f(x_{i+1}))$. Therefore, we can approximate

$$\int_a^bf(x)\,\text{d}x\approx\frac12\sum_{i=0}^{n-1}(x_{i+1}-x_i)(f(x_i)+f(x_{i+1}))$$

When all the trapezoids have the same width, or $x_{i+1}-x_i=\frac{b-a}n$, it is

$$\int_a^bf(x)\,\text{d}x\approx\frac{b-a}{2n}\left[f(x_0)+2f(x_1)+2f(x_2)+\dots+2f(x_{n-1})+f(x_n)\right]$$

where $x_0=a$, $x_1=a+\frac{b-a}n$, $x_2=a+2\frac{b-a}n$, $x_3=a+3\frac{b-a}n$, $\dots$, $x_{n-1}=a+(n-1)\frac{b-a}n$, and $x_n=b$.

The Trapezoidal Rule (or trapezoid rule) is a method to approximate the definite integral. The way this works is by "approximating" the area under a curve as a trapezoid. Therefore, the formula for the area under the curve (approximation) is $\int_{a}^{b} f(x)\, dx \approx (b-a) \left[\frac{f(a) + f(b)}{2} \right]$.

This rule is one of the Newton-Cotes formula.

Errors in the Method

If the integrand is concave up, this method will overestimate the true value of the area beneath the curve.