Compound Trapezoidal Method for Numerical Integration

Trapezoidal Rule:

The function f(x) (in blue) is approximated by a linear function (in red).

In mathematics, and more specifically in numerical analysis, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral.

\int _{a}^{b}f(x)\,dx

The trapezoidal rule works by approximating the region under the graph of the function $f(x)$ as a trapezoid and calculating its area. It follows that

$\int _{a}^{b}f(x)\,dx\approx (b-a)\cdot {\tfrac {f(a)+f(b)}{2}}$

The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way.

Illustration of ‘chained trapezoidal rule’ used on an irregularly-spaced partition of [a, b]. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this ‘chained’ (or ‘composite’) trapezoidal rule is usually what is meant by ‘integrating with the trapezoidal rule’. Let {xk} be a partition of [a, b] such that a = x0 < x1 < … < xn-1 < xn = b and dxk be the length of the k-th subinterval (that is, dxk = xk – xk-1), then

\int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\frac {f(x_{k-1})+f(x_{k})}{2}}\Delta x_{k}={\tfrac {\Delta x}{2}}\left(f(x_{0})+2f(x_{1})+2f(x_{2})+2f(x_{3})+2f(x_{4})+\cdots +2f(x_{n-1})+f(x_{n})\right)

The approximation becomes more accurate as the resolution of the partition increases (that is, for larger N, dxk decreases). When the partition has a regular spacing, as is often the case, the formula can be simplified for calculation efficiency.