08 Mar 2026
Numerical Integration: Trapezoid and Simpson Rules
Approximating definite integrals using equally spaced data and common quadrature formulas.
msc
semester-i
numerical-methods
integration
Numerical integration estimates a definite integral when the antiderivative is difficult or when the function is known only through tabulated data.
\[I=\int_a^b f(x)\,dx.\]Step size
Divide the interval $[a,b]$ into $n$ equal parts:
\[h=\frac{b-a}{n}.\]The points are
\[x_i=a+ih,\qquad i=0,1,2,\ldots,n.\]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.
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.
Discussion