Trapezoidal rule

In calculus, the trapezoidal rule (informally trapezoid rule; or in British English trapezium rule) is a technique for numerical integration, i.e. approximating the definite integral: $${\displaystyle \int _{a}^{b}f(x)\,dx.}$$

The trapezoidal rule works by approximating the region under the graph of the function ${\displaystyle f(x)}$ as a trapezoid and calculating its area. This is easily calculated by noting that the area of the region is made up of a rectangle with width ${\displaystyle (b-a)}$ and height ${\displaystyle f(a)}$, and a triangle of width ${\displaystyle (b-a)}$ and height ${\displaystyle f(b)-f(a)}$.

Letting ${\displaystyle A_{r}}$ denote the area of the rectangle and ${\displaystyle A_{\text{t}}}$ the area of the triangle, it follows that $${\displaystyle A_{\text{r}}=(b-a)\cdot f(a),\quad A_{\text{t}}={\tfrac {1}{2}}(b-a)\cdot [f(b)-f(a)].}$$

Therefore, $${\displaystyle {\begin{aligned}\int _{a}^{b}f(x)\,dx&\approx A_{\text{r}}+A_{\text{t}}\\&=(b-a)\cdot f(a)+{\tfrac {1}{2}}(b-a)\cdot [f(b)-f(a)]\\&=(b-a)\cdot \left(f(a)+{\tfrac {1}{2}}f(b)-{\tfrac {1}{2}}f(a)\right)\\&=(b-a)\cdot \left({\tfrac {1}{2}}f(a)+{\tfrac {1}{2}}f(b)\right)\\&=(b-a)\cdot {\tfrac {1}{2}}[f(a)+f(b)].\end{aligned}}}$$

The rule can also be derived by replacing the integrand with the equation of the line joining points ${\displaystyle {\big (}a,f(a){\big )}}$ and ${\displaystyle {\big (}b,f(b){\big )}}$, which using the two point form of the equation of a line, is $${\displaystyle y=(x-a)\,{\frac {f(b)-f(a)}{b-a}}+f(a).}$$

Therefore, $${\displaystyle {\begin{aligned}\int _{a}^{b}f(x)\,dx&\approx \int _{a}^{b}(x-a)\,{\frac {f(b)-f(a)}{b-a}}+f(a)\,dx\\&=\left[{\frac {1}{2}}(x-a)^{2}\,{\frac {f(b)-f(a)}{b-a}}+xf(a)\right]_{x=a}^{x=b}\\&=\left[{\frac {1}{2}}(b-a)^{2}\,{\frac {f(b)-f(a)}{b-a}}+bf(a)\right]-af(a)\\&={\frac {1}{2}}(b-a)[f(b)-f(a)]+(b-a)f(a)\\&={\frac {1}{2}}(b-a)[f(a)+f(b)],\end{aligned}}}$$ as before.

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 ${\displaystyle \{x_{k}\}}$ be a partition of ${\displaystyle [a,b]}$ such that ${\displaystyle a=x_{0}<x_{1}<\cdots <x_{N-1}<x_{N}=b,}$ and ${\displaystyle \Delta x_{k}}$ be the length of the ${\displaystyle k}$-th subinterval (that is, ${\displaystyle \Delta x_{k}=x_{k}-x_{k-1}}$), then $${\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\frac {f(x_{k-1})+f(x_{k})}{2}}\Delta x_{k}.}$$ The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums and is sometimes defined this way.

The approximation becomes more accurate as the resolution of the partition increases (that is, for larger ${\displaystyle N}$, all ${\displaystyle \Delta x_{k}}$ decrease).