Rules for extended intervals

Once given a rule for computing integrals $I_i$ over narrow intervals, Eq. (3) prescribes a method for numerical calculation of integrals over extended regions: for the midpoint rule,

$$M \equiv \sum_{i=1}^N M_i$$ (11)

$$= \sum f(x^{(m)}_i) h,$$

$$= \sum f\left(\frac{x_{i-1}+x_i}{2}\right) h,$$

and for the trapezoid rule,

$$T \equiv \sum_{i=1}^N T_i$$ (12)

$$= \sum_{i=1}^N \frac{f(x_{i-1})+f(x_i)}{2} h$$

$$= h \left( \begin{array}{ccccccccccc} \frac{f_0}{2} & + & \frac{f_1... ...\ & & & & & & & + & \frac{f_{N-1}}{2} & + & \frac{f_{N}}{2} \end{array}\right)$$

$$= h \left( \frac{1}{2} f_0 + f_1 +f_2 + \ldots + f_{N-1} + \frac{1}{2} f_{N} \right),$$

where $f_i\equiv f(x_i)$.