Error Terms for the Trapezoid, Midpoint, and Simpson\u27s Rules

When it is not possible to integrate a function we resort to Numerical Integration. For example the ubiquitous Normal curve tables are obtained using Numerical Integration. The antiderivative of the defining function for the normal curve involves the formula for antiderivative of e-x^2 which can\u27t be expressed in the terms of basic functions. Simpson\u27s rule is studied in most Calculus books, and in all undergraduate Numerical Analysis books, but proofs are not provided. Hence if one is interested in a proof of Simpson\u27s rule, either it can be found in advanced Numerical Analysis books as a special case of the so called Newton-Cotes formulas, or in math journals such as American Mathematical Monthly. My thesis adviser Hajrudin Fejzic, has recently published yet another proof. In this thesis I plan to introduce Numerical Integration formulas such as simpler Composite and Midpoint rules as well as Simpson\u27s rule and I will provide the proofs to these rules using the ideas developed in Dr. Fejzic\u27s publication as well as new proofs based on ideas of Dr. Fejzic that were communicated to me