11 An introduction to Riemann Integration


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1 11 An introduction to Riemnn Integrtion The PROOFS of the stndrd lemms nd theorems concerning the Riemnn Integrl re NEB, nd you will not be sked to reproduce proofs of these in full in the exmintion in Jnury However, you ARE expected to know the definitions nd the sttements of the results, nd to know how to pply these results. In prticulr, the EXAMPLES discussed in lectures ARE exminble. 1
2 11.1 Integrtion nd ntidifferentition From the modules G11CAL nd G11ACF you will be fmilir with integrtion s form of ntidifferentition, nd s re under the curve. There re some problems here. Does it mke sense to tlk bout re for complicted regions in R 2? Which functions hve ntiderivtives? An ntiderivtive for function f is differentible function F which stisfies F = f. Antiderivtives re lso clled primitives or indefinite integrls. 2
3 Recll tht the chrcteristic function of set E, χ E, is defined by χ E (x) = 1 if x E while χ E (x) = 0 if x E. On Question Sheet 5 there is n exmple of function which hs no ntiderivtive: the chrcteristic function of set with just one point. This shows tht some cre is needed! One of the min results of this chpter is the (first) Fundmentl Theorem of Clculus, one impliction of which is tht every continuous, relvlued function on n intervl hs n ntiderivtive. Of course, some discontinuous functions do hve ntiderivtives (cn you think of n exmple?). 3
4 11.2 Prtitions, res nd Riemnn sums Let nd b be rel numbers with < b. For bounded, relvlued function f defined on [, b], we now discuss prtitions P of [, b] nd the corresponding Riemnn upper sum nd Riemnn lower sum for f on [, b] (denoted by U(P, f) nd L(P, f) respectively). The bounded function f nd the intervl [, b] will be fixed throughout the following definitions. Definition A prtition of [, b] is finite set of points P = {x 0, x 1,..., x n } [, b], where = x 0 < x 1 < < x n = b. The points x 0, x 1,..., x n re clled the vertices of P. Gp to fill in 4
5 For 1 k n, we set M k (P, f) = sup{f(t) x k 1 t x k } nd m k (P, f) = inf{f(t) x k 1 t x k }. Gp to fill in 5
6 The Riemnn upper sum for f corresponding to P, U(P, f), nd the Riemnn lower sum for f corresponding to P, L(P, f), re defined by nd n U(P, f) = M k (P, f)(x k x k 1 ) k=1 n L(P, f) = m k (P, f)(x k x k 1 ). k=1 Gp to fill in 6
7 It is obvious tht we lwys hve L(P, f) U(P, f). With bit more work, we cn prove the following fct (see books for detils). Lemm Let P nd Q be prtitions of [, b]. Then L(P, f) U(Q, f). In words, this tells us the following. The Riemnn lower sum for f corresponding to prtition P of [, b] cn not be greter thn the Riemnn upper sum for f corresponding to prtition Q of [, b], even if P nd Q re different. 7
8 11.3 The Riemnn integrl With f nd [, b] s bove, the preceding lemm llows us to define the Riemnn lower nd upper integrls of f over the intervl [, b]. Definition The Riemnn lower integrl of f over the intervl [, b], f(x) dx nd the Riemnn upper integrl of f over the intervl [, b], f(x) dx re defined by f(x) dx = sup{l(p, f) : P is prtition of [, b] } nd f(x) dx = inf{u(q, f) : Q is prtition of [, b] }. 8
9 This in turn llows us to define Riemnn integrbility for f. Definition The bounded function f is Riemnn integrble on [, b] if f(x) dx = f(x) dx, i.e., if the Riemnn lower integrl is equl to the Riemnn upper integrl. In this cse we define the Riemnn integrl of f from to b to be the common vlue: f(x) dx = f(x) dx = f(x) dx. UNBOUNDED functions on n intervl [, b] re declred NOT to be Riemnn integrble. However they my hve improper integrls (s discussed in G11ACF). 9
10 Note the following fcts for bounded, relvlued functions on n intervl. The Riemnn lower integrl is lwys less thn or equl to the Riemnn upper integrl. Every Riemnn lower sum is less thn or equl to the lower integrl. Every Riemnn upper sum is greter thn or equl to the upper integrl. It is esy to show tht constnt functions re Riemnn integrble, with the obvious integrl (exercise). We will see below tht the fmily of Riemnnintegrble functions is firly lrge. However, not ll bounded functions re Riemnn integrble. 10
11 Exmple. Let f be the restriction of the chrcteristic function of the rtionls, χ Q, to [0, 1]. So f : [0, 1] R is defined by f(x) = 1 for x [0, 1] Q, while f(x) = 0 for x [0, 1] Q c. This bounded function f is not Riemnn integrble on [0, 1]. Gp to fill in 11
12 However, continuous functions re wellbehved. Theorem Every continuous, relvlued function on n intervl [, b] is Riemnn integrble on [, b]. The converse to this theorem is flse. There re mny discontinuous functions which re Riemnn integrble. For exmple (see Question Sheet 5), the chrcteristic function of singlepoint set is discontinuous, but is nevertheless Riemnn integrble. Gp to fill in 12
13 The Riemnn integrl behves s you expect sensible notion of integrtion to behve. Theorem Let f, g : [, b] R be Riemnn integrble nd λ R. Then f + g, λf nd f re lso Riemnn integrble nd the following hold. () (f(x) + g(x)) dx = (dditivity). f(x) dx + g(x) dx (b) λf(x) dx = λ f(x) dx. (c) If f(x) g(x) for ll x [, b], then f(x) dx g(x) dx. 13
14 (d) We lwys hve f(x) dx f(x) dx. (e) For ny c ], b[, we hve tht f is lso Riemnn integrble on [, c] nd on [c, b] nd f(x) dx = c f(x) dx + c f(x) dx. We now come to the finl (nd min) result of this chpter. 14
15 Theorem (Fundmentl Theorem of Clculus, lso known s the First Fundmentl Theorem of Clculus) Let f be continuous, relvlued function on [, b]. For x [, b] define F (x) = x f(t) dt. Then F is continuous on [, b], nd is differentible on ], b[, with F (x) = f(x) for ll x ], b[. From this it follows esily tht continuous, relvlued functions on intervls lwys hve ntiderivtives. It lso shows tht ntidifferentition is the correct wy to integrte continuous functions. See Question Sheet 5 for more detils. 15
16 As concrete exmple, consider the function G : ]0, [ R defined by G(x) = x 0 sin( t 3 ) dt. Then G is differentible on ]0, [, nd, for x ]0, [, we hve G (x) = sin( x 3 ). There is more powerful integrtion theory due to Henri Lebesgue. In this theory, χ Q is integrble on [0, 1] (with 1 0 χ Q(x) dx = 0), nd so re mny other strnge functions. The Lebesgue integrl is beyond the scope of this module, but it is n importnt tool in more dvnced nlysis nd in probbility theory. THE END Hve good holidy! 16
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