Anonymous From: - Posts: - Votes: - |
Define the definite (Riemann) integral for a continuous function, f, from a to b. |

Anonymous From: - Posts: - Votes: - |
The length of an interval [a, b] on the real line is defined to be b-a. Let P : a = x0 < x1 < ? < xk-1 < xk < ? < xp-1 < xp = b denote a partitioning of the interval [a, b] in to subintervals Sk = [xk-1, xk], 1 ? k ? p. Let xk Î Sk be arbitrary and let Dk = xk-xk-1 denote the length of the interval Sk. Let f(x) be a bounded real valued function defined on the interval [a, b]. The sum S(P) = S1? k? p f(xk)Dk, is called a Riemann sum of f(x) associated with the partition P. Let mk denotes the inf of f(x) on Sk, i.e., the greatest lower bound of f(x) on Sk, and, |

Anonymous From: - Posts: - Votes: - |
Let [a..b] be a closed interval of the set R of real numbers. Let f:R?R be a real function. Let f(x) be bounded on [a..b]. Suppose that ?y?R such that: For any lower sum L(P) over any of subdivision P of [a..b], L(P)=y For any upper sum U(P) over any of subdivision P of [a..b], U(P)=y Then y is known as the definite integral of f(x) over [a..b] and is denoted: y=?baf(x) dx |

TinyTeacher From: US Posts: 12 Votes: 12 |
F(x)=1/(b-a) a<x<b riemann integral=integ f(x) dx a<x<b =1 |

Anonymous From: - Posts: - Votes: - |
The length of an interval [a, b] on the real line is defined to be b-a. Let P : a = x0 < x1 < � < xk-1 < xk < � < xp-1 < xp = b denote a partitioning of the interval [a, b] in to subintervals Sk = [xk-1, xk], 1 � k � p. Let xk Î Sk be arbitrary and let Dk = xk-xk-1denote the length of the interval Sk. Let f(x) be a bounded real valued function defined on the interval [a, b]. The sum S(P) = S1� k� p f(xk)Dk, is called a Riemann sum of f(x) associated with the partition P. Let mk denotes the inf of f(x) on Sk, i.e., the greatest lower bound of f(x) on Sk, and, Mk denotes sup of f(x) on Sk, i.e., the least upper bound of f(x) on Sk. The lower Riemann sum and theupper Riemann sum of f(x) for the partition P are defined, respectively, as L(P) = S1� k� p mkDk, U(P) = S1� k� p MkDk, where mk denotes the inf of f(x) on Sk and Mk denotes sup of f(x) on Sk. Clearly, L(P) � U(P). The lower Riemann integral L (f), and the upper Riemann integral U (f) are defined by L (f) = sup {L(P)}, U (f) = inf {U(P)}, the sup and the inf running over all partitions P of [a, b]. Note that for all partitions P: L(P) � L (f), U (f) � U(P). Let Q be another partition. The partition R = PÈQ is obtained by taking its subintervals to be the intersections of subintervals of P with those of Q. Thus each subinterval in R is a subset of some subinterval in P as well as some subinterval in Q. Such an R is called a refinement of P. Since inf over a larger set is not larger, and the sup over a larger set is not smaller, it is clear that L(P) � L(PÈQ) � U(PÈQ) � U(Q). Taking sup over P�s, L (f) � U(Q). Next taking inf over Q�s, we get L(f) � U(f). Hence, for any partition P, L(P) � L(f) � U(f) � U(P). The function f(x) is called Riemann integrable over [a, b] if L (f) = U (f), and then this common value is defined to be the Riemann integral or the definite integral of f(x) written as ò(a,b) f(x)dx. The Riemann criterion for the Riemann integrability is: Theorem. f(x) is Riemann integrable iff for each e > 0, there exists a partition P such that U(P) - L(P) < e . Proof: The necessity follows from the definition of sup and inf. For sufficiency, 0 � U (f) - L (f) � U(P) � L(P) < e . Since e > 0 is arbitrary, by letting it to tend to zero it follows that U (f) = L (f). # For a partition P : a = x0 < � < xk-1 < xk < � < xp-1 < xp = b, the partition size is defined by D(P) = max1� k� p D k. Theorem. Let f(x) be Riemann integrable on [a, b], and let {S(Pi)} be a sequence of Riemann sums with limi® � D(Pi) = 0. Then limi®� S(Pi) = ò(a,b) f(x)dx. Proof: Let e > 0 be arbitrary. By the Riemann criterion, there exists a partition P such that U(P)�L(P) < e . Let Qi be the union of the partitions Pi and P. Let m = inf f(x) and M = sup f(x). Then L(P) � L(Qi) � U(Qi) � U(P), U(Pi)-U(Qi) � pD(Pi)(M-m), and L(Qi)�L(Pi) � pD(Pi)(M-m). Hence, U(Pi)-L(Pi) � U(Qi)�L(Qi)+2pD(Pi)(M-m). It follows that, limsupn® � [U(Pi)-L(Pi)] � U(Qi)�L(Qi) � U(P)�L(P) < e . Since e > 0 is arbitrary, it follows that limn®� [U(Pi)-L(Pi)] = 0. Hence, limi®� U(Pi) =ò(a,b) f(x)dx] = limi®� [L(Pi), so that by the sandwitch theorem, also, limi®� S(Pi) = ò(a,b) f(x)dx. # Two useful classes of Riemann integrable functions are as follows. Define the partition size D(Pn) = max1� k� n Dk. Theorem. If f(x) is continuous on [a, b], f(x) is Riemann integrable on [a, b]. Proof: Since f(x) is uniformly continuous on [a, b] given any e > 0, there exists a d > 0 such that |f(x)-f(y)| < e , for all |x-y| < d . Hence for any partition P such that D(P) < d , U(P) � L(P) < e(b-a), and the Riemann criterion is satisfied. Hece f(x) is Riemann integrable on [a, b]. # Theorem. If f(x) is monotone on [a, b], f(x) is Riemann integrable on [a, b]. Proof: For definiteness we assume that f(x) is non-decreasing, i.e., x � y implies f(x) � f(y). Then, for any partition P, U(P)-L(P) = S 1� k� p-1 [f(xk)-f(xk-1)]Dk � D(P)[f(b)-f(a)] and so taking D(P) sufficiently small the Riemann criterion is satisfied. # Area Under a Curve The area of a rectangle [a, b]�[c, d] is defined to be the product of its sides, namely, (b-a)�(d-c). The area under a curve C : y = f(x) � 0, a � x � b, where y(x) is a continuous function on [a, b] means the area of the region R bounded by C, x-axis and the vertical lines x = a, and x = b. To motivate the definition of this area, consider a partition Pn of [a, b]. Since, È1�k�n [xk-1, xk]�[0, mk] Ì R Ì È1�k�n [xk-1, xk]�[0, Mk] and the areas of the unions of rectangles are respectively L(Pn) and U(Pn) we should have L(Pn) � A � U(Pn) for all Pn. Hence, as U(Pn) and L(Pn) both converge to ò(a,b) f(x)dx, we arrive at: A = ò(a,b) f(x)dx. Hence for a region defined by R12 = {(x, y) : y1(x) � y � y2(x), a � x � b} the area becomes: A = ò(a,b) [y2(x)- y1(x)]dx. The area of a region that can be divided in to a finite numbers of sub-regions of type R12 can thus be obtained as the sum of the areas of the individual sub-regions. For more general regions the areas could be found by a limiting process involving series and improper integrals. s. |

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