Numerical Integration - The Trapezoidal Rule

• Given the function, $f(x)$, the antiderivative is a function $F(x)$ such that $F'(x)=f(x)$.
• The definite integral
  $$\int_a^b f(x)dx=F(b) - F(a)$$
  can be evaluated from the antiderivative.
• Still, there are functions that do not have an antiderivative expressible in terms of ordinary functions. Such as the function: $f(x)=e^x/log(x)$

      1 from sympy import *
      2 x = symbols('x')
      3 f = exp(x)/log(x)
      4 df = integrate(f, x) # Function derivative in symbolic form
      5 print(df)
      6 # Integral(exp(x)/log(x), x)
  

• Is there any way that the definite integral can be found when the antiderivative is unknown?
• We can do it numerically by using the composite trapezoidal rule
• The definite integral is the area between the curve of $f(x)$ and the $x$-axis.
• That is the principle behind all numerical integration;

  Figure 4.6: The trapezoidal rule.

  

  • We divide the distance from $x=a$ to $x=b$ into vertical strips and add the areas of these strips.
  • The strips are often made equal in widths but that is not always required.
  • Approximate the curve with a sequence of straight lines.
  • In effect, we slope the top of the strips to match with the curve as best we can.
• The gives us the trapezoidal rule. Figure illustrates this.

• It is clear that the area of the strip from $x_i$ to $x_{i+1}$ gives an approximation to the area under the curve:
  $$\int_{x_i}^{x_{i+1}}f(x)dx\approx\frac{f_i+f_{i+1}}{2}(x_{i+1}-x_i)$$

• We will usually write $h =(x_{i+1}-x_i)$ for the width of the interval.
• Error term for the trapezoidal integration is
  $$Error=-(1/12)h^3f''(\xi)=O(h^3)$$

• What happens, if we are getting the integral of a known function over a larger span of $x$-values, say, from $x=a$ to $x=b$?
• We subdivide [a,b] into $n$ smaller intervals with $\Delta x=h$, apply the rule to each subinterval, and add.