The Trapezoidal Rule

Figure 1: The trapezoidal rule.

• 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.
• We are approximating the curve with interpolating polynomials of degree-1.
• The gives us the trapezoidal rule. Figure 1 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 $x=b$?