Assignment:
Numerical approximations to integrals typically get better -- i.e., their error goes down -- proportional to a power of N, the number of subintervals in the interval of integration. For the upper and lower sums, the error typically goes down like 1/N as N increases. For the midpoint and trapezoidal rules, the error typically goes down like 1/N^2. For Simpson's rule, the error typically goes down like 1/N^4.
(a) Demonstrate this behavior numerically, using the integral of x^3 on [0,1] as a typical integral. (b) Demonstrate that this normal behavior is not seen in the integral of sqrt(x) on [0,1]! Apparently the slightly bad behavior of sqrt(x) at 0 (it is not differentiable there) is to blame.
Provide complete and step by step solution for the question and show calculations and use formulas.