RIEMANN INTEGRATION. Throughout our discussion of Riemann integration. B = B [a; b] = B ([a; b] ; R)

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1 RIEMANN INTEGRATION Throughout our disussion of Riemnn integrtion B = B [; b] = B ([; b] ; R) is the set of ll bounded rel-vlued funtons on lose, bounded, nondegenerte intervl [; b] : 1. DEF. A nite set of points P = fx 0 ; x 1 ; :::; x n g is prtition of [; b] if We sometimes write = x 0 < x 1 < < x n = b: P = f = x 0 < x 1 < < x n = bg : The set of ll prtitions of [; b] is denoted by P: 2. DEF. A prtition Q of [; b] is re nement of prtition P of [; b] if P Q: Then Q is sid to be ner thn P: 3. DEF. Let f 2 B [; b] nd P = fx 0 ; x 1 ; :::; x n g be prtition of [; b] : Then the i-th subintervl of the prtition is nd I i = [x i 1 ; x i ] ; ji i j = x i x i 1 m i = m i (f) = inf f (x) x2i i M i = M i (f) = sup f (x) x2i i Then nx L P (f) = m i ji i j is the lower sum of f with respet to the prtition P nd nx U P (f) = M i ji i j is the upper sum of f with respet to the prtition P: Fts: 1

2 1. For ny prtition P, L P (f) U P (f) : 2. If Q re nes P; then L P (f) L Q (f) nd U Q (f) U P (f) 3. If P nd Q re ny two prtitions of [; b] ; then L P (f) U Q (f) : 4. DEF. Let f 2 B [; b]. The lower integl of f with respet to the prtition P is (L) = L (f) = sup L P (f) P nd the upper integl of f with respet to the prtition P is (U) = U (f) = inf P U P (f) 5. Ft: (L) (U) L (f) U (f) 6. DEF. Let f 2 B [; b]. Then f is Riemnn integrble (R-integrble) on (over) [; b] if L (f) = U (f) in whih se the Riemnn integl of f on (over) [; b] is = L (f) = U (f) : Nottion: R [; b] is the set of ll R-integrble funtions on [; b] : 7. Fts: 2

3 1. Let 2 R: The funtion f : [; b]! R de ned by f (x) = is integrble over [; b] nd dx = (b ) 2. (Dirihlet Funtion) The funtion f de ned by 1 for x 2 [; b] \ Q f (x) = 0 for x 2 [; b] \ I is not Riemnn integrble over [; b] : Existene of the Integrl 8. Th. Let f 2 B [; b]. Then f is Riemnn integrble on [; b] if nd only if for every " > 0 there exists prtition P suh tht U P (f) L P (f) < ": 9. Th. If f : [; b]! R is monotone, then f is integrble on [; b] : 10. Th. If f : [; b]! R is ontinuous on [; b] ; then f is integrble on [; b] : 11. (Intervl Additivity) Let [ 0 ; b 0 ] [; b] nd 2 [; b] : Then f 2 R [; b] =) f 2 R [ 0 ; b 0 ] ; f 2 R [; ] nd f 2 R [; b] =) f 2 R [; b] nd Z + = Note: A diret onsequene of 9, 10, nd 11 is: If f is de ned on n intervl [; b] tht n be deomposed into djent subintervls on eh of whih f is either monotone or ontinuous, then f is integrble over the full intervl [; b]. 12. Th. (Order Properties) If f 2 R [; b] nd f 0 on [; b] ; then 0: 3

4 Cor. If f; g 2 R [; b] nd f g on [; b] ; then g (x) dx: Cor. If f 2 R [; b] nd m f M on [; b] ; then m (b ) M (b ) : Combintions of Integrble Funtions 13. (Linerity) If f; g 2 R [; b] nd 2 R; then f + g 2 R [; b] nd f 2 R [; b] : 14. If f; g 2 R [; b] then f 2 2 R [; b] fg 2 R [; b] 15. (Tringle Inequlity for Integrls) If f 2 R [; b] then jfj 2 R [; b] nd jf (x)j dx 16. (Intermedite Vlue Theoem for Integrls) If f 2 C [; b] nd g 2 R [; b] is positive on [; b] ; then there is point 2 [; b] suh tht f (x) g (x) dx = f () g (x) dx: Cor: If f 2 C [; b] ; then there is point 2 [; b] suh tht = f () (b ) : 4

5 Interlude 17. DEF: In writing f 2 R [; b] it ws ssumed tht < b: Then by de nition nd Z = b Z = 0: Remrks: 1. These equlities beome theorems if we djust our previous de nitions so tht prtition P = f = x 0 ; x 1 ; :::; x n = bg where the intermedite points in the prtition inrese if < b nd derese if b < : When = b there is only one prtition whih is f = x 0 ; b = x n g. 2. With these extensions the order properties only hold when < b: 3. The other results hold for ny nd b nd intervl dditivity is true for ny order of the points ; b; nd : 4. It is lso useful to note tht regrdless of the order of nd b; jf (x)j dx The Fundmentl Theorem (FTC) of Clulus Throughout this topi f : I! R is funtion de ned on n intervl of ny type nd f 2 R [; b] for every losed, bounded subintervl [; b] I: Also, 2 I is xed nd F : I! R is the funtion de ned by F (x) = Z x f (t) dt: 18. If f : I! R is s desribed, then F (x) = R x f (t) dt is ontinuous on I: 19. (FTC I) If f : I! R is s desribed, then F (x) = R x f (t) dt is di erentible t eh point x 2 I t whih f is ontinuous nd F 0 (x) = f (x) : 5

6 20. (FTC II) If f : [; b]! R is ontinuous nd F is ny ntiderivtive of f on I; then = F (b) F () : 21. (FTC III) If f 0 2 R [; b] ; then f 0 (x) dx = f (b) f () : Convergene nd the Integrl 22. Th. Th. If f n 2 R [; b] nd f n onverges uniformly to f on [; b] ; then f 2 R [; b] nd s n! 1; equivlently, lim n!1 f n (x) dx! f n (x) dx = lim f n (x) dx: n!1 The Riemnn Sum Connetion 23. DEF. Let P = f = x 0 ; x 1 ; :::; x n = bg be prtition of [; b] nd i 2 I i : Then nx f ( i ) ji i j is Riemnn sum of f over (on) [; b] : 23. Th. Let f 2 B [; b] : Then f is R-integrble over [; b] if nd only if there is rel number A suh tht for every " > 0 there is prtition P of [; b] suh tht n A X f ( i ) ji i j < " for ll hoies of i 2 I i : 6