The Composite Trapezoidal Rule

• This gives the composite trapezoidal rule;
  $$\boxed{\int_a^b\approx\sum_{i=0}^{n-1} \frac{h}{2} (f_i+f_{i+1})=\frac{h}{2}(f_0+2f_1+2f_2+\ldots+2f_{n-1}+f_n)}$$

• The error is not the local error $O(h^3)$ but the global error, the sum of $n$ local errors;
  $$Global~error= \frac{-(b-a)}{12}h^2f''(\xi)=O(h^2)$$
  where $nh=(b - a)$
• Use the trapezoidal rule to estimate the integral from $x=1.8$ to $x=3.4$ for $f(x)=e^x$.
• Applying the trapezoidal rule:
  

Apply The trapezoidal rule to $f(x)=e^x$ in interval of 1.8 and 3.4. (Example py-file: mytrapezoid.py)

Figure 4.7: Integration for $f(x)=e^x$ by the trapezoidal rule.

Apply The trapezoidal rule to Simple Pendulum to find the period. (Example py-file: simplependulum.py)

Table 4.4: Integration for $\frac {1.0}{\sqrt {1.0-(sin(\theta _0/2)sin(\phi ))^2}}$ by the trapezoidal rule.