Week 10: Riemann integral and its properties


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1 Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07
2 Motivtion: Computing flow from flow rtes 1 We observe the flow of wter through drin, which vries with time. The result is flow rte, in litres/second, continuously recorded over time intervl [, b]. From these dt, we wnt to determine the totl mount A of wter tht hs pssed through the vlve during the intervl. This vlue corresponds to the re under the grph of f
3 Answer for constnt rte 2 If the flow is constnt, sy equl to c, the nswer is esily obtined: A = (b ) c This corresponds to the formul re = width height. for rectngulr res. The ide to clculte the re under rbitrry grphs is to pproximte the grph by piecewise constnt functions.
4 Are under the grph: Piecewise constnt functions 3 A piecewise constnt function or step function is function f : [, b] R tht consists of finitely mny constnt pieces Here, the region under the grph is mde up out of rectngles nd its re is computed by summing the res of the rectngles.
5 Prtition 4 Definition. Let I = [, b] R be some intervl. A prtition of I is given by finite subset P = {x 0,..., x n } stisfying {, b} P. Without loss of generlity, = x 0 < x 1 < x 2 <... < x n = b. Exmple: The set P = {0, 0.3, 0.5, 0.8, 1.0} defines prtition of the intervl [0, 1].
6 Approximtion by step functions 5 Definition. Let f : [, b] R be function, nd P = {x 0, x 1,..., x n } prtition. We define M k (f) = sup{f(x) : x k < x < x k+1 } M k (f) = inf{f(x) : x k < x < x k+1 } Interprettion: M k nd M k provide optiml pproximtion of the grph of f by step functions with jumps in P, one from bove, one from below.
7 Exmple: Approximtion from bove 6 A function defined on [0, 3], prtition P = {0, 1, 2, 3}. Blue: Function grph, Blck: Step function ssocited to M k
8 Exmple: Approximtion from below 7 A function defined on [0, 3], prtition P = {0, 1, 2, 3}. Blue: Function grph, Blck: Step function ssocited to M k
9 Upper nd lower sum 8 Definition. Let f : [, b] R, nd let P = {x 0, x 1,..., x n } be prtition of [, b], with = x 0 < x 1 <... < x n = b.. We write n S(P) = M k 1 (x k x k 1 ) Interprettion: S(P) = k=1 n M k 1 (x k x k 1 ) k=1 The re below the step function with vlues M k 1 contins the re below the grph of f. Hence S(P) is greter or equl to the re below the grph of f. Likewise: S(P) is smller or equl to the re below the grph of f.
10 Grphicl interprettion of upper nd lower sum 9 The difference S(P) S(P) is the re between upper nd lower step function pproximtion
11 Refinement of prtition 10 Definition. Let P 1, P 2 be two prtitions of [, b]. Then P 1 is clled refinement of P 2 if P 1 P 2. Interprettion: If P 1 P 2, then S(P 2 ) S(P 1 ) S(P 1 ) S(P 2 ) Hence the re between upper nd lower pproximtion decreses. The two should pproximte the sme vlue, s the prtition gets finer nd finer.
12 Illustrtion for refinement 11 A function f : [0, 3] R, prtition {0, 1, 2, 3}, lower nd upper pproximtion
13 Illustrtion for refinement 12 The sme function, lower nd upper pproximtion for the refinement {0, 0.5, 0.7, 0.8, 0.9, 1, 1.3, 1.5, 1.6, 1.7, 1.8, 1.9, 2, 2.2, 2.4, 2.6, 2.8, 3}.
14 Riemnn integrble function 13 Definition. The function f : [, b] R is clled (Riemnn) integrble if for every ɛ > 0 there is prtition P of [, b] such tht S(P) S(P) < ɛ Note: This implies for every refinement P of P. S(P ) S(P ) < ɛ
15 Convergence of upper nd lower sums 14 Theorem 1. Let f be Riemnn integrble function. Let P n be sequence of prtitions stisfying δ n 0, where δ n is the mximl distnce of two neighboring elements of P n. Then I(f) = lim n S(P n ) exists, with I(f) = lim n S(P n ). Moreover, I(f) is the sme for ll sequences of prtitions with δ n 0.
16 Definition of the Riemnn integrl 15 Definition. If f is integrble, I(f) s in Theorem 1. I(f) is clled the (Riemnn) integrl of f over [, b], nd denoted s b f(x)dx. is clled lower bound of the integrl, b is clled upper bound of the integrl, nd f is clled the integrnd. Furthermore, we define, for < b, s well s b f(x)dx = b f(x)dx = 0 f(x)dx
17 Criteri for Riemnn integrbility 16 Sufficient conditions: If f is continuous on [, b], then f is integrble. If f is monotonic nd bounded on [, b], then f is integrble. Exmple: A bounded function tht is not integrble: f : [0, 1] R, f(x) = { 1 x Q 1 x Q For every prtition P, one finds S(P) = 1 1 = S(P).
18 Properties of the Riemnn integrl 17 Theorem 2. Let f, g be integrble over the intervl with bounds, b, let s R sf is integrble, with b sf(x)dx = s b f(x)dx. f + g is integrble, with b f(x) + g(x)dx = b f(x)dx + b g(x)dx. Let c in R be such tht f is integrble over [b, c]. Then f is integrble over [, c], with c f(x)dx = b f(x)dx + c b f(x)dx. If f is integrble, then f is integrble s well, with b b f(x)dx f(x) dx
19 Monotonicity of integrls 18 Theorem 3. Let b, let f : [, b] R be integrble nd bounded, with Then m f(x) M, for ll x [, b] m(b ) b f(x)dx M(b ). This pplies in prticulr, when f is continuous on [, b], nd m = min f(x), M = mx f(x). x [,b] x [,b] More generlly, if f, g : [, b] R re integrble, with f(x) g(x) for ll x [, b], then b f(x)dx b g(x)dx.
20 Illustrtion for the estimte 19
21 Fundmentl Theorem of Clculus 20 Theorem 4. Let f : [, b] R be continuous. We define F : [, b] R, F (y) = y f(x)dx Then F is continuous on [, b], differentible on (, b), with F (x) = f(x), x (, b). Conversely, suppose tht G : [, b] R is continuous, differentible on (, b) with G = f. Then the integrl is computed s b f(x)dx = G b := G(b) G()
22 Integrtion nd ntiderivtives 21 Remrks: Let f : [, b] R be continuous function. A differentible function F with F = f is clled ntiderivtive or primitive of f. Hence f hs primitive given by F (y) = y f(x)dx. Two primitives F, G of f only differ by constnt: F (x) = G(x) c, with c R fixed. By letting F (y) = y f(x)dx one obtins the unique primitive of f stisfying F () = 0. It is customry to denote primitives s F = f(x)dx (without bounds), nd refer to them s indefinite integrls of f.
23 Appliction: The length of curve 22 Definition. Let f : [, b] R n be given, i.e., The set f(x) = (f 1 (x), f 2 (x),..., f n (x)) T. C = {f(x) : x [, b]} is clled curve in R n, nd f is clled prmeteriztion of C. We ssume tht ll f i re continuously differentible on (, b) nd continuous on [, b]. We define the length of C s l(c) = b f 1 (x) 2 + f 2 (x) f n(x) 2 dx
24 Exmple: Circumference of the circle 23 We consider the mp f : [0, 2π] R 2, with f(x) = (sin(x), cos(x)). The resulting curve is the unit circle. We compute f 1(x) = cos(x), f 2(x) = sin(x) nd thus, using sin 2 + cos 2 = 1, 2π 2π f 1 (x) 2 + f 2 (x)2 dx = 1dx = 2π. 0 0
25 Exmple: Length of grph 24 We wnt to determine the length of the grph G f of f(t) = t 2, for t [0, 1]. G f is prmeterized by g : [0, 1] R 2, g(t) = (t, t 2 ) T. Using g 1(t) = 1, g 2(t) = 2t, we obtin One cn check tht F (t) = 1 4 l(g f ) = t2 dt. ( 2t 1 + 4t 2 + ln(2t + ) 1 + 4t 2 )) is primitive of g(t) = 1 + 4t 2. Hence, l(g f ) = F 1 0 = 1 ( ln(2 + ) 5) 0 4
26 Summry 25 Definition nd interprettion of integrls; re under the grph Properties of the integrl: Linerity, monotonicity Evlution of integrls vi ntiderivtives ( New problem: How to obtin ntiderivtives) Appliction of integrls: Curve length
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