Summary of Numerical Integration

We can use numerical integration to estimate the values of definite integrals when a closed form of the integral is difficult to find or when an approximate value only of the definite integral is needed.
The most commonly used techniques for numerical integration are the midpoint rule, trapezoidal rule, and Simpson’s rule.
The midpoint rule approximates the definite integral using rectangular regions whereas the trapezoidal rule approximates the definite integral using trapezoidal approximations.
Simpson’s rule approximates the definite integral by first approximating the original function using piecewise quadratic functions.

Key Equations

Midpoint rule ${M}_{n}=\displaystyle\sum _{i=1}^{n}f\left({m}_{i}\right)\Delta x$
Trapezoidal rule ${T}_{n}=\frac{1}{2}\Delta x\left(f\left({x}_{0}\right)+2f\left({x}_{1}\right)+2f\left({x}_{2}\right)+\cdots +2f\left({x}_{n - 1}\right)+f\left({x}_{n}\right)\right)$
Simpson’s rule ${S}_{n}=\frac{\Delta x}{3}\left(f\left({x}_{0}\right)+4f\left({x}_{1}\right)+2f\left({x}_{2}\right)+4f\left({x}_{3}\right)+2f\left({x}_{4}\right)+4f\left({x}_{5}\right)+\cdots +2f\left({x}_{n - 2}\right)+4f\left({x}_{n - 1}\right)+f\left({x}_{n}\right)\right)$
Error bound for midpoint rule $\text{Error in }{M}_{n}\le \frac{M{\left(b-a\right)}^{3}}{24{n}^{2}}$
Error bound for trapezoidal rule $\text{Error in }{T}_{n}\le \frac{M{\left(b-a\right)}^{3}}{12{n}^{2}}$
Error bound for Simpson’s rule $\text{Error in }{S}_{n}\le \frac{M{\left(b-a\right)}^{5}}{180{n}^{4}}$

absolute error: if $B$ is an estimate of some quantity having an actual value of $A$ , then the absolute error is given by $|A-B|$

midpoint rule: a rule that uses a Riemann sum of the form ${M}_{n}=\displaystyle\sum _{i=1}^{n}f\left({m}_{i}\right)\Delta x$ , where ${m}_{i}$ is the midpoint of the ith subinterval to approximate ${\displaystyle\int }_{a}^{b}f\left(x\right)dx$

numerical integration: the variety of numerical methods used to estimate the value of a definite integral, including the midpoint rule, trapezoidal rule, and Simpson’s rule

relative error: error as a percentage of the absolute value, given by $|\frac{A-B}{A}|=|\frac{A-B}{A}|\cdot 100\text{%}$

Simpson’s rule: a rule that approximates ${\displaystyle\int }_{a}^{b}f\left(x\right)dx$ using the integrals of a piecewise quadratic function. The approximation ${S}_{n}$ to ${\displaystyle\int }_{a}^{b}f\left(x\right)dx$ is given by ${S}_{n}=\frac{\Delta x}{3}\left(\begin{array}{c}f\left({x}_{0}\right)+4f\left({x}_{1}\right)+2f\left({x}_{2}\right)+4f\left({x}_{3}\right)+2f\left({x}_{4}\right)+4f\left({x}_{5}\right)\\ +\cdots +2f\left({x}_{n - 2}\right)+4f\left({x}_{n - 1}\right)+f\left({x}_{n}\right)\end{array}\right)$ trapezoidal rule a rule that approximates ${\displaystyle\int }_{a}^{b}f\left(x\right)dx$ using trapezoids