08 Mar 2026
Numerical Integration: Trapezoid and Simpson Rules
Approximating definite integrals using equally spaced data and common quadrature formulas.
The definite integral represents accumulated quantity: area under a curve, work done by a variable force, charge from a current-time graph, or probability over an interval. If the antiderivative is not known, or if the function is available only as tabulated data, the integral is estimated numerically.
\[I=\int_a^b f(x)\,dx.\]The basic idea is to divide the interval into small parts and replace the curve by simple shapes whose areas can be calculated.
Step size
Divide the interval $[a,b]$ into $n$ equal parts:
\[h=\frac{b-a}{n}.\]The points are
\[x_i=a+ih,\qquad i=0,1,2,\ldots,n.\]The values $y_i=f(x_i)$ are called ordinates. Most elementary formulas assume equal spacing.
Trapezoidal rule
The trapezoidal rule approximates the area by trapezia:
\[\int_a^b f(x)\,dx \approx \frac{h}{2}\left[ y_0+y_n+2\sum_{i=1}^{n-1}y_i \right].\]It is simple and works well when the function is nearly linear over each small interval.
Simpsonās one-third rule
For an even number of intervals $n$, Simpsonās rule is
\[\int_a^b f(x)\,dx \approx \frac{h}{3} \left[ y_0+y_n +4\sum_{\text{odd }i}y_i +2\sum_{\substack{\text{even }i\\i\ne 0,n}}y_i \right].\]Simpsonās rule uses parabolic approximation and is usually more accurate than the trapezoidal rule for smooth functions.
Comparison on a simple integral
Evaluate
\[I=\int_0^1 x^2\,dx\]using $h=0.5$. The tabulated values are
| $x$ | $y=x^2$ |
|---|---|
| 0 | 0 |
| 0.5 | 0.25 |
| 1.0 | 1.00 |
By the trapezoidal rule,
\[I_T=\frac{0.5}{2}\,[0+1+2(0.25)] =0.375.\]By Simpsonās one-third rule,
\[I_S=\frac{0.5}{3}\,[0+1+4(0.25)] =\frac{1}{3}.\]The exact integral is
\[\int_0^1x^2\,dx=\frac13.\]Thus Simpsonās rule gives the exact answer here because it is exact for polynomials up to degree three.
Key points
- Smaller step size generally improves accuracy.
- Simpsonās rule requires an even number of intervals.
- Numerical integration should be repeated with a smaller $h$ to check convergence.
Practice questions
- State the trapezoidal rule for equally spaced ordinates.
- State Simpsonās one-third rule and mention its condition on $n$.
- Evaluate $\int_0^1x^2\,dx$ using the trapezoidal rule with $h=0.5$.
- Evaluate the same integral using Simpsonās one-third rule.
- Why is step-size reduction used as a check in numerical integration?
Discussion