Trapezoidal and Simpson rules

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

For an approximation use numerical quadrature. Trapezoids use segments; Simpson uses parabolas and requires even n. Check smoothness, singularities and error: an approximation is not an exact primitive.

Step-by-step example

1. Apply Simpson with two subintervals.

   $$\int_0^1x^2\,dx,\quad h=1/2$$

2. Evaluate at 0, 1/2 and 1.

   $$S_2=\frac16[0+4(1/4)+1]$$

3. Here it gives the exact value because the function is quadratic.

   $$S_2=1/3$$