Numerical Integration

The basic problem is to obtain an approximation to the value of a definite integral of the form

$$\int_{a}^{b}$$

where a and b are finite.

Most methods use the approach of dividing the interval [a, b] into a large number N of subdivisions. We will denote the i^th subdivision point by x_i (x₀ = a and x_N = b). We write y_i = f (x_i). If all subdivisions are of equal width h then x_i = a + ih, and h = (b - a)/N.

Trapezium Rule:  This method uses an arbitrary number of equal subdivisions.

$$\approx {\frac{h}{2}}$$

The error I - I_N is roughly proportional to h² so that doubling the number of strips should roughly quarter the error.

Simpson's Rule:  This method uses any even number 2N of equal subdivisions. Note that in this formula h = (b - a)/2N.

$$\approx {\frac{h}{3}}$$

The error I - I_2N is roughly proportional to h⁴.

Simpson's Rule is far more accurate that the Trapezium Rule, in general, for essentially the same amount of calculation.