Summary of Numerical Integration

Essential Concepts

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

Glossary

absolute error
if [latex]B[/latex] is an estimate of some quantity having an actual value of [latex]A[/latex], then the absolute error is given by [latex]|A-B|[/latex]
midpoint rule
a rule that uses a Riemann sum of the form [latex]{M}_{n}=\displaystyle\sum _{i=1}^{n}f\left({m}_{i}\right)\Delta x[/latex], where [latex]{m}_{i}[/latex] is the midpoint of the ith subinterval to approximate [latex]{\displaystyle\int }_{a}^{b}f\left(x\right)dx[/latex]
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 [latex]|\frac{A-B}{A}|=|\frac{A-B}{A}|\cdot 100\text{%}[/latex]
Simpson’s rule
a rule that approximates [latex]{\displaystyle\int }_{a}^{b}f\left(x\right)dx[/latex] using the integrals of a piecewise quadratic function. The approximation [latex]{S}_{n}[/latex] to [latex]{\displaystyle\int }_{a}^{b}f\left(x\right)dx[/latex] is given by [latex]{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)[/latex] trapezoidal rule a rule that approximates [latex]{\displaystyle\int }_{a}^{b}f\left(x\right)dx[/latex] using trapezoids