Today we reviewed the Midpoint Rule and covered the Trapezoidal and Simpson’s Rule.
The Trapezoidal Rule replaces rectangle for each sub-interval by a trapezoid whose area equals the average of the left-endpoint and right-endpoint Riemann rectangles:
$$ t_i = (l_i + r_i)/2 = \frac{\triangle x}{2} \left( f(x_{i-1}) + f(x_{i}) \right) $$
The Simpson’s Rule uses a parabola instead, but since a parabola is defined by 3 coefficients, 3 pieces of data and so 2 intervals are need to fit a parabola. We demonstrated briefly why for each pair of sub-intervals \( [x_{2i-2},x_{2i}]\), \( i=1,2,…,n/2\), we have the approximating area under parabolas:
$$ s_i = \frac{\triangle x}{3} \left( f(x_{2i-2}+4f(x_{2i-1})+f(x_{2i}) \right) $$
The scan is here: wk6day12. The extra typed note demonstrating the order of convergence and error bounds are here: wk6day12_errbd.
We accidentally covered also some basic examples of improper integrals, with two major types.
- Type I: “infinite” horizontal stretch, either \( a = -\infty\) or \( b = \infty\), or both.
- Type II: “infinite” vertical stretch, with the interval of integration [a,b] having a or b as singular points of the integrand. Actually, one could allow singular points c, to be in the interior of the interval too, i.e. \( a<c<b\)
We will cover these cases in detail on Thursday. The basic technique is to apply techniques of definite integrals and use limits.