It is often the case, when evaluating definite integrals, that an antiderivative for the integrand cannot be found, or is extremely difficult to find. In some instances, a numerical approximation to the value of the definite value will suffice. The following techniques can be used, and are listed in rough order of ascending complexity.
A special case of the Riemann sum, where we let , in other words the point on the far right-side of each sub-interval on, . Again if we pick n to be finite, then we have:
Another special case of the Riemann sum, this time we let , which is the point on the far left side of each sub-interval on . As always, this is an approximation when is finite. Thus, we have: