Numerical Integration Methods
- 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.
Improper Integrals
- Integrals of functions over infinite intervals are defined in terms of limits.
- Integrals of functions over an interval for which the function has a discontinuity at an endpoint may be defined in terms of limits.
- The convergence or divergence of an improper integral may be determined by comparing it with the value of an improper integral for which the convergence or divergence is known.
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] - Improper integrals
[latex]\begin{array}{c}{\displaystyle\int }_{a}^{+\infty }f\left(x\right)dx=\underset{t\to \text{+}\infty }{\text{lim}}{\displaystyle\int }_{a}^{t}f\left(x\right)dx\hfill \\ {\displaystyle\int }_{\text{-}\infty }^{b}f\left(x\right)dx=\underset{t\to \text{-}\infty }{\text{lim}}{\displaystyle\int }_{t}^{b}f\left(x\right)dx\hfill \\ {\displaystyle\int }_{\text{-}\infty }^{+\infty }f\left(x\right)dx={\displaystyle\int }_{\text{-}\infty }^{0}f\left(x\right)dx+{\displaystyle\int }_{0}^{+\infty }f\left(x\right)dx\hfill \end{array}[/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]
- improper integral
- an integral over an infinite interval or an integral of a function containing an infinite discontinuity on the interval; an improper integral is defined in terms of a limit. The improper integral converges if this limit is a finite real number; otherwise, the improper integral diverges
- 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