Trapezoidal Rule

Topics: Numerical Methods

The Trapezoidal Rule is a numerical integration technique used to approximate the value of an indefinite integral or the area under a curve.

It consists of dividing the full integration interval $[a,b]$ into sub-intervals, connecting the top points of each interval and summing the areas of the obtained trapezoids.

The sub-intervals do not have to be equal. In this case, the general formula is
$\int_a^b f(x) dx \approx \sum_{k=1}^N \frac{f(x_{k-1})+f(x_k)}{2} \; \Delta x$
In the case of equal spacing, the formula can be simplified:
$\int_a^b f(x)dx \approx \Delta x \Bigg[ \frac{f(x_0) + f(x_N)}{2} + \sum_{k=1}^{N-1}f(x_k) \Bigg]$
For a running integral, the trapezoidal formula is
$I_n = I_{n-1} + \frac{h}{2} \Big[f(x_{n-1}) + f(x_n) \Big]$

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