Real Analysis/Riemann integration

../ Riemann integration Applications of Derivatives Fundamental Theorem of Calculus __TOC__ Definition Riemann integration is the formulation of integration most people think of if they ever think about integration. It is the only type of integration considered in most calculus classes; many other forms of integration, notably Lebesgue integrals, are extensions of Riemann integrals to larger classes of functions. The Riemann integral was developed by Bernhard Riemann in 1854 and was, when invented, the first rigorous definition of integration applicable to not necessarily continuous functions. We will first define some preliminary ideas. Partitions Definition Let A Partition is defined as the ordered -tuple of real numbers such that Norm of a Partition Let be a partition given by Then, the Norm (or the "mesh") of is defined as Tagged Partition Let be a partition A Tagged Partition is defined as the set of ordered pairs such that . The points are called Tags. Riemann thumb|180px|Riemann sum of a function Riemann Sums Let Let be a tagged partition of The Riemann Sum of over with respect to is given by Riemann Integral Let Let We say that is Integrable on if and only if, for every there exists such that for every partition satisfying , we have that is said to be the integral of over , and is written as or as Properties Theorem (Uniqueness) Let be integrable on Then the integral of is unique Proof Assume, if possible that are both integrals of over . Consider As are integrals, there exist such that for all that satisfy and for all that satisfy Let . Hence, if is a partition satisfying , then we have and that That is, , which is an obvious contradiction. Hence the integral of is unique. We now state (without proof) two seemingly obvious properties of the integral. Theorem Let be integrable and let Then: (i) (ii) Theorem (Boundedness Theorem) Let be Riemann integrable. Then is bounded over Proof Assume if possible that is unbounded. For every divide the interval into parts. Hence, for every , is unbounded on at least one of these parts. Call it . Now, let be given. Consider an arbitrary . Let be a tagged partition such that and , where is taken so as to satisfy . Thus we have that . But as is arbitrary, we have a contradiction to the fact that is Riemann integrable. Hence, is bounded. Integrability We now study classes of Riemann integrable functions. The first "constraint" on Riemann integrable functions is provided by the Cauchy Integrability Criterion. Theorem (Cauchy Criterion) Let Then, (i) is Riemann integrable on if and only if (ii) For every , there exists such that if are two partitions satisfying then Proof ()Let and let be given. Then, there exists such that for every partition satisfying ,we have Now, let partitions be such that . Thus we have that , that is () For every , consider such that for all partitions satisfying , we have . Without loss of generality, we can assume that when . For every , let be a partition such that The sequence is a Cauchy sequence, and hence it has a limit . Now, for every , we have a such that implies . Thus Theorem (Squeeze Theorem) Let Then, (i) is Riemann integrable on if and only if (ii) For every , there exist Riemann integrable functions such that for all and Proof ()Take . It is easy to see that ()Let . Then, there exist functions such that . Further, if and , then there exist such that if a partition satisfies then and then Now let be a partition satisfying . Now, we can easily see that . Hence, is a Cauchy sequence, with a limit , and as in the previous proof, we can show that