The Riemann-Stieltjes Integral

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "The Riemann-Stieltjes Integral"

Transcription

1 Chpter 6 The Riemnn-Stieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0 < 1 < 2 < < n 1 < n = b. We write i = i i 1 for i = 1, 2,, n, nd define the norm P of P by P = m 1 i n i. (b A refinement of prtition P is prtition Q suh tht P Q. In this se, we lso sy tht Q is finer thn P. Given two prtitions P 1 nd P 2 of [, b], we ll the union P 1 P 2 their ommon refinement. Definition 6.2. Let α be monotonilly inresing funtion on [, b], f be bounded funtion on [, b] nd P = { 0,..., n } be prtition of [, b]. Define nd M j (f = We ll the numbers α j = α( j α( j 1 (1 j n sup f(, m j (f = inf f( [ j 1, j ] [ j 1, j ] U(P, f, α = M j (f α j, L(P, f, α = (1 j n. m j (f α j the upper (Riemnn-Stieltjes sum nd, respetively, the lower (Riemnn-Stieltjes sum of f with prtition P over [, b] with respet to α. Note tht, if m f( M for ll [, b], then m m j (f M j (f M for eh j = 1, 2,, n, nd hene m(α(b α( L(P, f, α U(P, f, α M(α(b α(. 1

2 2 6. The Riemnn-Stieltjes Integrl So, the sets {U(P, f, α P is prtition of [, b]} nd {L(P, f, α P is prtition of [, b]} re bounded sets in R. nd Define fdα = inf{u(p, f, α P is prtition of [, b]} fdα = sup{l(p, f, α P is prtition of [, b]} to be the upper Riemnn-Stieltjes integrl nd, respetively, the lower Riemnn- Stieltjes integrl of f over [, b] with respet to α. We sy tht f is Riemnn-Stieltjes integrble on [, b] with respet to α, nd write f R(α[, b], provided tht (6.1 fdα = fdα. In this se, the ommon vlue of the upper nd lower Riemnn-Stieltjes integrls in (6.1 is lled the Riemnn-Stieltjes integrl of f over [, b] with respet to α nd denoted by Sometimes, we lso write fdα = fdα. f(dα(; of ourse, the dummy vrible n be repled by ny other letters (eept for the letters f, α or d, to void obvious onfusion. Definition 6.3. When the funtion α is the identity funtion, i.e., α( =, we define the nottions U(P, f, L(P, f, to be the nottions, respetively, U(P, f, α, L(P, f, α, fd, fdα, fd, fdα, fd, R[, b] fdα, R(α[, b]. In this se, if f R[, b], then we sy f is Riemnn integrble on [, b] or, simply, integrble on [, b]. Theorem 6.1. If P, Q re prtitions of [, b] nd Q is finer thn P, then L(P, f, α L(Q, f, α U(Q, f, α U(P, f, α. Tht is, L(P, f, α inreses with P nd U(P, f, α dereses with P. Proof. Sine Q is obtined from P by dding finitely mny points, by indution, we only need to prove the se when Q is obtined from P by dding one etr point. So let P = { 0, 1,..., k 1, k,..., n }, Q = { 0, 1,..., k 1, y, k,..., n }, where k 1 < y < k. Then L(P, f, α = m j (f α j, where m j (f = inf [ j 1, j ]

3 6.1. Definition nd Eistene of the Integrl 3 Note tht k 1 L(Q, f, α = ( m j (f α j + inf f( [ k 1,y] ( + inf f( (α( k α(y + [y, k ] inf f( inf f(, [ k 1,y] [ k 1, k ] j=k+1 (α(y α( k 1 m j (f α j. inf f( inf f(. [y, k ] [ k 1, k ] Hene, sine α is inresing, we hve tht ( ( inf [ k 1,y] (α(y α( k 1 + inf [y, k ] (α( k α(y ( inf [ k 1, k ] (α(y α( k 1 + α( k α(y = m k (f α k. Consequently, k 1 L(Q, f, α m j (f α j + m k (f α k + j=k+1 m j (f α j = L(P, f, α. The proof of U(Q, f, α U(P, f, α is similr. Theorem 6.2. fdα fdα. Proof. Let P, Q be rbitrry two prtitions of [, b]. Then, sine P Q is refinement of both P nd Q, by the previous theorem, Hene L(P, f, α L(P Q, f, α U(P Q, f, α U(Q, f, α. fdα = sup{l(p, f, α} inf{u(q, f, α} = P Q fdα. Theorem 6.3 (Criterion for Integrbility. A bounded funtion f is in R(α[, b] if nd only if for eh ε > 0 there eists prtition P of [, b] suh tht (6.2 U(P, f, α L(P, f, α < ε. Proof. (Suffiieny for Integrbility. Let ε > 0. Assume tht there eists prtition P of [, b] suh tht U(P, f, α L(P, f, α < ε. Then U(P, f, α < L(P, f, α + ε, nd thus fdα U(P, f, α < L(P, f, α + ε fdα + ε. Sine ε > 0 is rbitrry, this proves tht fdα fdα, nd hene fdα = fdα; so f R(α[, b]. (Neessity for Integrbility. Assume f R(α[, b]; nmely, fdα = fdα. Let ε > 0. Then there eist prtitions P 1, P 2 of [, b] suh tht U(P 1, f, α < fdα + ε/2, L(P 2, f, α > fdα ε/2.

4 4 6. The Riemnn-Stieltjes Integrl Let P = P 1 P 2 be the ommon refinement of P 1 nd P 2. Then P is prtition of [, b], nd, using fdα = fdα, we hve ( U(P, f, α L(P, f, α U(P 1, f, α L(P 2, f, α < fdα + ε ( fdα ε = ε. 2 2 Theorem 6.4. Suppose tht (6.2 holds for prtition P = { 0, 1,, n }. ( Then (6.2 holds with P repled by ny refinement of P. (b If s j, t j re points in [ j 1, j ] for eh j, then f(s j f(t j α j < ε. ( If f R(α[, b], then f(t j α j for ll t j [ j 1, j ] with j = 1, 2,, n. fdα < ε Proof. ( follows esily sine L(P, f, α inreses with P nd U(P, f, α dereses with P. (b follows sine both f(s j, f(t j re between m j (f nd M j (f nd hene f(s j f(t j M j (f m j (f. The obvious inequlities L(P, f, α f(t j α j U(P, f, α, L(P, f, α fdα U(P, f, α prove (. Theorem 6.5. If f is ontinuous on [, b], then f R(α[, b]. Proof. Let ε > 0 be given, nd hoose η > 0 so tht (α(b α(η < ε. Sine f is uniformly ontinuous on [, b], there eists δ > 0 suh tht f( f(y < η for ll, y [, b] with y < δ. Let P = { 0,..., n } be ny prtition of [, b] with norm P < δ. Sine f is ontinuous on eh subintervl [ j 1, j ], by the Etreme Vlue Theorem, there eist j, d j [ j 1, j ] suh tht M j (f = f( j, m j (f = f(d j. Sine j d j j P < δ, we hve Therefore, U(P, f, α L(P, f, α = M j (f m j (f = f( j f(d j < η (M j (f m j (f α j η So, by the Criterion for Integrbility, f R(α[, b]. (1 j n. α j = η(α(b α( < ε. Theorem 6.6. If α is ontinuous on [, b], then every monotoni funtion on [, b] belongs to R(α[, b].

5 6.1. Definition nd Eistene of the Integrl 5 Proof. Without loss of generlity, we ssume f is monotonilly inresing funtion on [, b]. Let ε > 0 be given, nd hoose η > 0 so tht (f(b f(η < ε. Sine α is uniformly ontinuous on [, b], there eists δ > 0 suh tht α( α(y < η for ll, y [, b] with y < δ. Let P = { 0,..., n } be ny prtition of [, b] with norm P < δ. Sine f is monotonilly inresing on eh subintervl [ j 1, j ], we hve Sine j j 1 = j P < δ, we hve m j (f = f( j 1, M j (f = f( j. α j = α( j α( j 1 < η (1 j n. Therefore, U(P, f, α L(P, f, α = (f( j f( j 1 α j η (f( j f( j 1 = η(f(b f( < ε. So, by the Criterion for Integrbility, f R(α[, b]. Theorem 6.7. Suppose f is bounded on [, b], f hs only finitely mny points of disontinuity on [, b], nd α is ontinuous t every point t whih f is disontinuous. Then f R(α[, b]. Proof. Let ε > 0 be given. Put M = sup [,b] f( nd let E be the set of points in [, b] t whih f is disontinuous. Sine E is finite nd α is ontinuous t every point of E, we n over E by finitely mny disjoint intervls [u j, v j ] in [, b] suh tht (α(v j α(u j < ε. Furthermore, we n hoose these intervls in suh wy tht every point of E (, b lies in the interior of some [u j, v j ]. If E, we ssume [, v 0 ] is one of these intervls; if b E, ssume [u 0, b] is one of these intervls. Remove ll open intervls (u j, v j, nd possibly [, v 0 or (u 0, b] if or b is in E. The remining set K is then ompt, nd f is ontinuous on K. Hene f is uniformly ontinuous on K, nd there eists δ > 0 suh tht f(s f(t < ε if s, t K nd s t < δ. Define prtition P = { 0, 1,, n } of [, b] s follows. All u j, v j belong to P. No points in (u j, v j belong to P, nd no points in [, v 0 or (u 0, b] belong to P. If i 1 is not one of u j, then i < δ. Note tht M i (f m i (f 2M for every i, nd tht M i (f m i (f ε unless i 1 is one of u j. Hene U(P, f, α L(P, f, α = (M i (f m i (f α i = i 1 =u j (M i (f m i (f α i + i=1 Sine ε > 0 is rbitrry, this proves f R(α[, b]. i 1 u j (M i (f m i (f α i 2Mε + ε(α(b α(. Theorem 6.8. Suppose f R(α[, b], m f M, φ is ontinuous on [m, M], nd h( = φ(f( on [, b]. Then h R(α[, b].

6 6 6. The Riemnn-Stieltjes Integrl Proof. Let ε > 0 be given. Sine φ is uniformly ontinuous on [m, M], there eists δ (0, ε suh tht φ(s φ(t < ε if s t δ nd t, s [m, M]. Sine f R(α[, b], there is prtition P = { 0, 1,, n } of [, b] suh tht (6.3 U(P, f, α L(P, f, α < δ 2. Let M j (f, m j (f be defined for f s bove; similrly, let M j (h, m j (h be defined for funtion h. Divide the numbers i = 1, 2,, n into two lsses: i A if M i (f m i (f < δ; i B if M i (f m i (f δ. For i A, we hve f( f(y M i (f m i (f < δ for ll, y [ i 1, i ]; hene, our hoie of δ shows tht M i (h m i (h = sup φ(f( [ i 1, i ] inf φ(f(y = y [ i 1, i ] For i B, we hve M i (h m i (h 2K, where K = δ i B sup (φ(f( φ(f(y ε.,y [ i 1, i ] sup φ(t < +. By (6.3, t [m,m] α i i B(M i (f m i (f α i U(P, f, α L(P, f, α < δ 2, so tht i B α i < δ. Thus it follows tht U(P, h, α L(P, h, α = i A (M i (h m i (h α i + i B(M i (h m i (h α i ε i A α i + 2K i B α i ε[α(b α(] + 2Kδ < ε[α(b α( + 2K]. Sine ε > 0 is rbitrry, by the Criterion of integrbility, this proves h R(α[, b] Further Properties of the Integrl Theorem 6.9. The Riemnn-Stieltjes integrl hs the following properties. (6.4 ( (Liner property If f 1, f 2 R(α[, b], then 1 f f 2 R(α[, b] for ll rel numbers 1, 2, nd ( 1 f f 2 dα = 1 f 1 dα + 2 f 2 dα. (b (Order property If f 1, f 2 R(α[, b] nd f 1 ( f 2 ( on [, b], then f 1 dα f 2 dα. ( (Additivity If f R(α[, b] nd < < b, then f R(α[, ] nd f R(α[, b]; moreover, fdα = fdα + fdα. Conversely, if < < b nd if f R(α[, ] nd f R(α[, b], then f R(α[, b], nd (6.4 holds. (d (Positive ombintion If f R(α 1 [, b] nd f R(α 2 [, b], nd k 1, k 2 re nonnegtive onstnts, then f R(k 1 α 1 + k 2 α 2 [, b], nd fd(k 1 α 1 + k 2 α 2 = k 1 fdα 1 + k 2 fdα 2.

7 6.2. Further Properties of the Integrl 7 (e (Absolute integrbility If f R(α[, b], then f R(α[, b], nd fdα f dα. Proof. Let s prove ( nd (e only. ( Assume f R(α[, b]. Let [, d] be subintervl of [, b]. Let ε > 0. Choose prtition P of [, b] suh tht U(P, f, α L(P, f, α < ε. Let P = P {, d} nd P 1 = P [, d]. Then P is refinement of P on [, b] nd P 1 is prtition of [, d], whih is prt of prtition P of [, b]. Therefore, we hve U d (P 1, f, α L d (P 1, f, α U(P, f, α L(P, f, α U(P, f, α L(P, f, α < ε, where U d (P 1, f, α, L d (P 1, f, α re for f defined on [, d], with P 1 being prtition on [, d]. Hene, by the Criterion for integrbility, f R(α[, d]. Now ssume < < b nd f R(α[, ] nd f R(α[, b]. Let ε > 0. prtitions P 1 of [, ] nd P 2 of [, b] suh tht U (P 1, f, α L (P 1, f, α < ε/2, Let P = P 1 P 2. Then P is prtition of [, b], nd we hve U b (P 2, f, α L b (P 2, f, α < ε/2. Choose U(P, f, α L(P, f, α = [U (P 1, f, α + U b (P 2, f, α] [L (P 1, f, α + L b (P 2, f, α] < ε. Hene, f R(α[, b]. To verify the dditivity property (6.4, ssume P is prtition of [, b]. Let P 0 = P {}, P 1 = P 0 [, ], nd P 2 = P 0 [, b]. Then P 0 = P 1 P 2 nd Hene U(P, f, α U(P 0, f, α = U (P 1, f, α + U b (P 2, f, α L(P, f, α L(P 0, f, α = L (P 1, f, α + L b (P 2, f, α fdα fdα + but, fdα = fdα, nd this proves fdα = fdα, fdα + fdα fdα. fdα + fdα + fdα + fdα; fdα, fdα. (e Assume f R(α[, b]. Then, with φ(t = t in Theorem 6.8, we hve f R(α[, b]. Sine f( f( f( ( [, b], by prt ( nd (b, we hve whih proves fdα f dα. f dα fdα f dα,

8 8 6. The Riemnn-Stieltjes Integrl Remrk 6.4. The onverse of (e in the theorem is flse. Indeed, onsider the funtion { 1 Q f( = 1 / Q. Then f( = 1 is onstnt nd hene f R[, b]; however, it is esily seen tht nd hene f / R[, b]. fd = b > 0, Theorem If f, g R(α[, b], then fg R(α[, b]. fd = b < 0, Proof. Using φ(t = t 2 in Theorem 6.8, we hve f 2, g 2, (f + g 2 R(α[, b], nd hene fg = (f + g2 f 2 g 2 2 R(α[, b]. Theorem Assume α is monotonilly inresing nd differentible on [, b] nd α R[, b]. Let f be bounded on [, b]. Then f R(α[, b] if nd only if fα R[, b]. In this se, fdα = f(α (d. Proof. Let ε > 0. Sine α R[, b], there is prtition P = { 0,, n } of [, b] suh tht (6.5 U(P, α L(P, α < ε. By the MVT, for eh j = 1, 2,..., n, there eists t j ( j 1, j suh tht α j = α (t j j. Let s j [ j 1, j ] be rbitrry. Then, by (6.5 nd Theorem 6.4, we hve α (t j α (s j j < ε. Put M = sup [,b] f < +. Sine α j = α (t j j, it follows tht f(s j α j f(s j α (s j j Mε. In prtiulr, f(s j α j U(P, fα + Mε; f(s j α (s j j U(P, f, α + Mε. Sine s j [ j 1, j ] is rbitrry, these two inequlities imply tht nd hene tht (6.6 U(P, f, α U(P, fα + Mε; fdα U(P, fα + Mε, U(P, fα U(P, f, α + Mε, fα d U(P, f, α + Mε.

9 6.2. Further Properties of the Integrl 9 Now note tht (6.5 remins true if P is repled by the refinement P Q, where Q is ny given prtition of [, b]. Hene (6.6 lso remins true with P repled by P Q, whih yields tht (6.7 fdα U(Q, fα + Mε, fα d U(Q, f, α + Mε for ll prtitions Q of [, b]. Tking the infim over prtitions Q, we hve fdα Sine ε > 0 is rbitrry, this implies fα d + Mε, fdα = fα d fα d, fdα + Mε. whih is vlid for ll bounded funtions f. The equlity fdα = fα d follows in etly the sme mnner. Hene the theorem is proved. Theorem 6.12 (Chnge of Vribles. Suppose φ is stritly inresing ontinuous from n intervl [A, B] onto [, b]. Suppose α is monotonilly inresing on [, b] nd f R(α[, b]. Define β nd g on [A, B] by β(y = α(φ(y, g(y = f(φ(y. Then g R(β[A, B], nd (6.8 B A gdβ = fdα. Proof. To eh prtition P = { 0,, n } of [, b] orresponds unique prtition Q = {y 0,, y n } of [A, B] suh tht j = φ(y j, nd vie vers. Then α j = α( j α( j 1 = α(φ(y j α(φ(y j 1 = β(y j β(y j 1 = β j for eh j = 1, 2,..., n. Sine the vlues tken by f on [ j 1, j ] re etly the sme s those tken by g on [y j 1, y j ], we see tht U(Q, g, β = U(P, f, α, L(Q, g, β = L(P, f, α. If f R(α[, b], then, for every ε > 0, it follows tht U(P, f, α L(P, f, α < ε for some prtition P of [, b]; hene, with the orresponding prtition Q of [A, B], we hve U(Q, g, β L(Q, g, β < ε. Hene g R(β[A, B], nd (6.8 holds. Combing the previous two theorems, we obtin the following hnge of vrible theorem for Riemnn integrls. Theorem If f R[, b] nd if φ: [A, B] [, b] is stritly inresing nd differentible with φ R[A, B], then where = φ(a, b = φ(b. f( d = B A f(φ(yφ (y dy,

10 10 6. The Riemnn-Stieltjes Integrl 6.3. Integrtion nd Differentition Theorem Let f R[, b]. Define F ( = f(tdt ( b. Then F is ontinuous on [, b]; furthermore, if f is ontinuous t point [, b], then F is differentible t, with F ( = f(. Proof. Suppose M = sup [,b] f < +. If < y b, then y F (y F ( = f(t dt M(y. Hene F is uniformly ontinuous on [, b]. Now ssume f is ontinuous t point [, b]. Given ε > 0, hoose δ > 0 suh tht f(t f( < ε t [, b], t < δ. Then, for ny [, b] with < < + δ, F ( F ( f( = 1 (f(t f( dt 1 nd similrly, for for ny [, b] with δ < <, F ( F ( f( = 1 (f(t f( dt 1 Hene, whenever [, b] nd 0 < < δ, it follows tht F ( F ( f( < ε; so F ( = f(. f(t f( dt < ε, f(t f( dt < ε. Theorem 6.15 (Fundmentl Theorem of Clulus. If f R[, b] nd if there is differentible funtion F on [, b] suh tht F = f, then f( d = F (b F (. Proof. Let ε > 0 be given. Choose prtition P = { 0,..., n } of [, b] suh tht U(P, f L(P, f < ε. The MVT furnishes points t j ( j 1, j suh tht Thus But hene F ( j F ( j 1 = f(t j j f(t j j = L(P, f (j = 1, 2,..., n. (F ( j F ( j 1 = F (b F (. f(t j j U(P, f, L(P, f f(t dt U(P, f; b F (b F ( f(t dt U(P, f L(P, f < ε. Sine ε > 0 is rbitrry, this ompletes the proof.

11 6.5. Retifible Curves in R k 11 Theorem 6.16 (Integrtion by Prts. Suppose F, G re differentible on [, b], F = f R[, b], nd G = g R[, b]. Then F (g( d = F (bg(b F (G( f(g( d. Proof. Clerly F g, fg R[, b]. Let H( = F (G( on [, b]. Then H ( = f(g(+ g(f ( nd hene H R[, b]. So, by Theorem 6.15, H(b H( = proving the theorem. H ( d = 6.4. Integrtion of Vetor-Vlued Funtions f(g( d + g(f ( d, Definition 6.5. Let f : [, b] R k nd α: [, b] R be monotonilly inresing on [, b]. Let f( = (f 1 (,, f k (, whih eh oordinte funtion f j is rel vlued. We sy f R(α[, b] if eh of its oordinte funtions f j R(α[, b]. In this se, we define ( f dα = f 1 dα,, f k dα. Mny of the results on rel-vlued funtions lso hold for these vetor-vlued funtions. To illustrte, we stte the nlogue of the fundmentl theorem of lulus. Theorem If f nd F mp [, b] into R k, f R[, b], nd F = f on [, b], then f(d = F(b F(. Theorem If f R(α[, b], then f R(α[, b], nd f dα f dα. Proof. If f = (f 1,, f k then f = (f f k 21/2. Hene if eh f j R[, b], then f R[, b]. To show the inequlity on the integrls, we ssume y = f dα 0; otherwise there is nothing to prove. By linerity nd order properties nd the Cuhy- Shwrz inequlity, y 2 = y f dα = Cnelling y > 0 proves the inequlity Retifible Curves in R k y f dα y f dα = y f dα. Definition 6.6. A ontinuous funtion γ from n intervl [, b] into R k is lled urve in R k. To emphsize the prmeter intervl [, b], we my lso sy tht γ is urve on [, b]. If γ is one-to-one, then γ is lled n r. If γ( = γ(b then γ is sid to be losed urve. Note tht urve in R k is funtion, not the rnge of γ, whih is point set in R k ; different urves my hve the sme rnge.

12 12 6. The Riemnn-Stieltjes Integrl Let γ : [, b] R k be urve. We ssoite to eh prtition P = { 0,, n } of [, b] the number Λ(P, γ = γ( j γ( j 1. (This ould be defined for urves in ny metri spe X, with γ( j γ( j 1 repled by d(γ( j, γ( j 1. Define the length of γ to be the number (inluding + Λ(γ = sup Λ(P, γ, where the supremum is tken over ll prtitions P of [, b]. We sy tht γ is retifible if Λ(γ < +. Theorem If γ is ontinuous on [, b], then γ is retifible, nd Λ(γ = γ (t dt. Proof. If j 1 < j b, then j γ( j 1 γ( j = γ (t dt j 1 Hene Λ(P, γ j j 1 γ (t dt = j j 1 γ (t dt. γ (t dt for ll prtitions P of [, b]. Consequently, Λ(γ γ (t dt. To show the opposite inequlity, let ε > 0 be given. Sine γ is uniformly ontinuous on [, b], there eists δ > 0 suh tht γ (t γ (s < ε if s t < δ, s, t [, b]. Let P = { 0,, n } be prtition of [, b] with P < δ. If t [ j 1, j ] then γ (t γ ( j + ε. Hene, by Theorem 6.17, j j γ (t dt γ ( j j + ε j = [γ (t + γ ( j γ (t] dt j 1 j 1 + ε j j j γ (t dt j 1 + [γ ( j γ (t] dt j 1 + ε j j γ( j γ( j 1 + γ ( j γ (t dt + ε j j 1 γ( j γ( j 1 + 2ε j. Adding these inequlities, we obtin γ (t dt Λ(P, γ + 2ε(b. Sine ε > 0 is rbitrry, it follows tht γ (t dt Λ(P, γ. This ompletes the proof. Suggested Homework Problems Pges Problems: 1 5, 7 9, 15, 17, 19

13 6.6. Improper Riemnn Integrls* Improper Riemnn Integrls* Definition 6.7. Let (, b R, where < b +, nd f : (, b R. We sy tht f is lolly integrble on (, b if f R[, d] for eh finite losed subintervl [, d] of (, b. We sy tht f is improperly (Riemnn integrble on (, b if f is lolly integrble on (, b nd the it ( (6.9 f( d = f( d + d b eists nd is finite. In this se, this it is lled the improper (Riemnn integrl of f on (, b. Sometimes we lso use the nottion f( d = + f( d to distinguish the improper integrls from the Riemnn integrls defined erlier. Lemm The order of its in (6.9 does not mtter. In prtiulr, if the it in (6.9 eists nd is finite, then the it ( f( d eists nd equls the it in (6.9. d b Proof. Let 0 (, b. Then ( f( d + d b + ( 0 = f( d + f( d + d b 0 0 (6.10 = f( d + + d b f( d. 0 Sine, for eh, 0 b ( d b d b f(d eists, we hve f( d 0 [ = 0 b d b = d b [ ( 0 ] = f( d f( d 0 b d b f( d 0 ] f( d f( d 0 b 0 Therefore, in (6.10 letting 0 b, we obtin tht ( 0 ( 0 b + f( d = + f( d = 0. d b f( d.

14 14 6. The Riemnn-Stieltjes Integrl Remrk 6.8. (i If f is integrble on [, b] for ll (, b, then the improper Riemnn integrl of f on (, b is lso given by f( d = f( d := f( d. + + If this it eists nd is finite, we lso sy tht f is improperly integrble on (, b]. The similr sitution pplies t the endpoint b, in whih se we sy tht f is improperly integrble on [, b. (ii It is esily seen tht f is improperly integrble on (, b if nd only if f is improperly integrble on (, ] nd on [, b for ll (, b. In this se, we hve tht + f( d = + f( d + f( d. Theorem The funtion f( = 1/ p is improperly integrble on (0, 1] if nd only if p < 1, nd is improperly integrble on [1, + if nd only if p > 1. Proof. Eerise! Theorem 6.22 (Liner Property. If f, g re improperly integrble on (, b nd k, l R, then kf + lg is improperly integrble on (, b, nd (kf( + lg( d = k f( d + l g( d. Proof. Use the Liner Property of integrls on eh subintervl [, d] of (, b. Theorem 6.23 (Comprison Theorem for Improper Integrls. Suppose tht f, g re lolly integrble on (, b nd 0 f( g( for ll (, b. If g is improperly integrble on (, b, then f is lso improperly integrble on (, b nd f( d g( d. Proof. Fi (, b. Let F (d = f( d nd G(d = g(d for d [, b. Then by the Order Property, F (d G(d. Note tht F nd G re inresing on [, b nd G(b eists. Hene F is bounded bove by G(b nd so F (d eists nd is finite. This shows tht f is improperly integrble on [, b. By the similr rgument, we lso show tht f is improperly integrble on (, ]; thus f is improperly integrble on (, b. The order property f( d g( d. follows esily from the order property of the Riemnn integrls of f nd g on eh subintervl [, d] of (, b. Emple 6.1. Show tht f( = (sin / 3/2 is improperly integrble on (0, 1]. Proof. Sine 0 sin for ll [0, 1] (use elementry lulus to prove it!, it follows tht 0 f( 3/2 = 1/2 (0, 1]. Sine 1/2 is improperly integrble on (0, 1], by the theorem bove, f is improperly integrble on (0, 1]. Emple 6.2. Show tht f( = (ln / 5/2 is improperly integrble on [1, +.

15 6.6. Improper Riemnn Integrls* 15 Proof. Sine 0 ln for ll 1 (use elementry lulus to prove it!, it follows tht 0 f( 5/2 = 3/2 1. Sine 3/2 is improperly integrble on [1, +, by the theorem bove, f is improperly integrble on [1, +. Lemm If f is bounded nd lolly integrble on (, b nd g is improperly integrble on (, b, then fg is improperly integrble on (, b. Proof. Use 0 fg M g nd the Comprison Theorem bove. Definition 6.9. Let f : (, b R. We sy tht f is bsolutely integrble on (, b if f is lolly integrble on (, b nd f is improperly integrble on (, b. We sy tht f is onditionlly integrble on (, b if f is improperly integrble on (, b but f is not improperly integrble on (, b. Theorem If f is bsolutely integrble on (, b, then f is improperly integrble on (, b nd f( d f( d. Proof. Sine 0 f + f 2 f, by the Comprison Theorem, f + f is improperly integrble on (, b. Hene, by the Liner Property, f = ( f + f f is lso improperly integrble on (, b. Moreover, for ll < d in (, b, f( d f( d. We then omplete the proof by tking the it s + nd d b. The onverse of Theorem 6.25 is flse. Emple 6.3. Prove tht f( = sin is onditionlly integrble on [1, +. Proof. Integrting by prts, we hve for ll d > 1, sin = os d d os 2 Sine 1/ 2 is bsolutely integrble on [1, +, we hve (os / 2 is bsolutely integrble on [1, + ; hene (os / 2 is improperly integrble on [1, +. Tking the it s d + bove, we hve + sin + os d = os( d eists nd is finite. This proves tht (sin / is improperly integrble on [1, +. We now show tht sin / is not improperly integrble on [1, +, whih proves tht (sin / is onditionlly integrble on [1, +. Note tht if n N nd n 2 then nπ sin kπ sin 1 kπ d d sin d = 2 1 kπ π k. 1 k=2 (k 1π k=2 Hene nπ sin d = +. n 1 So sin / is not improperly integrble on [1, +. (k 1π d. k=2

MATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals.

MATH 409 Advanced Calculus I Lecture 22: Improper Riemann integrals. MATH 409 Advned Clulus I Leture 22: Improper Riemnn integrls. Improper Riemnn integrl If funtion f : [,b] R is integrble on [,b], then the funtion F(x) = x f(t)dt is well defined nd ontinuous on [,b].

More information

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions. Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene

More information

(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.

(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P. Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time

More information

Week 7 Riemann Stieltjes Integration: Lectures 19-21

Week 7 Riemann Stieltjes Integration: Lectures 19-21 Week 7 Riemnn Stieltjes Integrtion: Lectures 19-21 Lecture 19 Throughout this section α will denote monotoniclly incresing function on n intervl [, b]. Let f be bounded function on [, b]. Let P = { = 0

More information

More Properties of the Riemann Integral

More Properties of the Riemann Integral More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl

More information

MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES

MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES CHARLIE COLLIER UNIVERSITY OF BATH These notes hve been typeset by Chrlie Collier nd re bsed on the leture notes by Adrin Hill nd Thoms Cottrell. These

More information

Math 554 Integration

Math 554 Integration Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we

More information

MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL

MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL DR. RITU AGARWAL MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR, INDIA-302017 Tble of Contents Contents Tble of Contents 1 1. Introduction 1 2. Prtition

More information

Riemann Stieltjes Integration - Definition and Existence of Integral

Riemann Stieltjes Integration - Definition and Existence of Integral - Definition nd Existence of Integrl Dr. Adity Kushik Directorte of Distnce Eduction Kurukshetr University, Kurukshetr Hryn 136119 Indi. Prtition Riemnn Stieltjes Sums Refinement Definition Given closed

More information

6.1 Definition of the Riemann Integral

6.1 Definition of the Riemann Integral 6 The Riemnn Integrl 6. Deinition o the Riemnn Integrl Deinition 6.. Given n intervl [, b] with < b, prtition P o [, b] is inite set o points {x, x,..., x n } [, b], lled grid points, suh tht x =, x n

More information

1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.

1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q. Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the

More information

MATH 409 Advanced Calculus I Lecture 18: Darboux sums. The Riemann integral.

MATH 409 Advanced Calculus I Lecture 18: Darboux sums. The Riemann integral. MATH 409 Advnced Clculus I Lecture 18: Drboux sums. The Riemnn integrl. Prtitions of n intervl Definition. A prtition of closed bounded intervl [, b] is finite subset P [,b] tht includes the endpoints

More information

ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs

ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Eletronis Tehnologil Edutionl Institute of Lmi, Greee EMil: {kehrin,

More information

Solutions to Assignment 1

Solutions to Assignment 1 MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove

More information

MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.

MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals. MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded

More information

Properties of the Riemann Stieltjes Integral

Properties of the Riemann Stieltjes Integral Properties of the Riemnn Stieltjes Integrl Theorem (Linerity Properties) Let < c < d < b nd A,B IR nd f,g,α,β : [,b] IR. () If f,g R(α) on [,b], then Af +Bg R(α) on [,b] nd [ ] b Af +Bg dα A +B (b) If

More information

Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.

Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer. Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points

More information

Chapter 4. Lebesgue Integration

Chapter 4. Lebesgue Integration 4.2. Lebesgue Integrtion 1 Chpter 4. Lebesgue Integrtion Section 4.2. Lebesgue Integrtion Note. Simple functions ply the sme role to Lebesgue integrls s step functions ply to Riemnn integrtion. Definition.

More information

Chapter 6. Riemann Integral

Chapter 6. Riemann Integral Introduction to Riemnn integrl Chpter 6. Riemnn Integrl Won-Kwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl

More information

6.5 Improper integrals

6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

f (x)dx = f(b) f(a). a b f (x)dx is the limit of sums

f (x)dx = f(b) f(a). a b f (x)dx is the limit of sums Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx

More information

Line Integrals and Entire Functions

Line Integrals and Entire Functions Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series

More information

Section 4.4. Green s Theorem

Section 4.4. Green s Theorem The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls

More information

Week 10: Riemann integral and its properties

Week 10: Riemann integral and its properties Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the

More information

Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx

Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.

More information

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1) Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

More information

Lecture 1 - Introduction and Basic Facts about PDEs

Lecture 1 - Introduction and Basic Facts about PDEs * 18.15 - Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1 - Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV

More information

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

More information

7.2 The Definition of the Riemann Integral. Outline

7.2 The Definition of the Riemann Integral. Outline 7.2 The Definition of the Riemnn Integrl Tom Lewis Fll Semester 2014 Upper nd lower sums Some importnt theorems Upper nd lower integrls The integrl Two importnt theorems on integrbility Outline Upper nd

More information

Introduction to Olympiad Inequalities

Introduction to Olympiad Inequalities Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................

More information

MATH 174A: PROBLEM SET 5. Suggested Solution

MATH 174A: PROBLEM SET 5. Suggested Solution MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion

More information

Euler-Maclaurin Summation Formula 1

Euler-Maclaurin Summation Formula 1 Jnury 9, Euler-Mclurin Summtion Formul Suppose tht f nd its derivtive re continuous functions on the closed intervl [, b]. Let ψ(x) {x}, where {x} x [x] is the frctionl prt of x. Lemm : If < b nd, b Z,

More information

For a continuous function f : [a; b]! R we wish to define the Riemann integral

For a continuous function f : [a; b]! R we wish to define the Riemann integral Supplementry Notes for MM509 Topology II 2. The Riemnn Integrl Andrew Swnn For continuous function f : [; b]! R we wish to define the Riemnn integrl R b f (x) dx nd estblish some of its properties. This

More information

Line Integrals. Partitioning the Curve. Estimating the Mass

Line Integrals. Partitioning the Curve. Estimating the Mass Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to

More information

Math 324 Course Notes: Brief description

Math 324 Course Notes: Brief description Brief description These re notes for Mth 324, n introductory course in Mesure nd Integrtion. Students re dvised to go through ll sections in detil nd ttempt ll problems. These notes will be modified nd

More information

Math 360: A primitive integral and elementary functions

Math 360: A primitive integral and elementary functions Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

More information

Global alignment. Genome Rearrangements Finding preserved genes. Lecture 18

Global alignment. Genome Rearrangements Finding preserved genes. Lecture 18 Computt onl Biology Leture 18 Genome Rerrngements Finding preserved genes We hve seen before how to rerrnge genome to obtin nother one bsed on: Reversls Knowledge of preserved bloks (or genes) Now we re

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

Calculus and linear algebra for biomedical engineering Week 11: The Riemann integral and its properties

Calculus and linear algebra for biomedical engineering Week 11: The Riemann integral and its properties Clculus nd liner lgebr for biomedicl engineering Week 11: The Riemnn integrl nd its properties Hrtmut Führ fuehr@mth.rwth-chen.de Lehrstuhl A für Mthemtik, RWTH Achen Jnury 9, 2009 Overview 1 Motivtion:

More information

Continuous Random Variables

Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

More information

Lecture 1. Functional series. Pointwise and uniform convergence.

Lecture 1. Functional series. Pointwise and uniform convergence. 1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

More information

Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction

Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued

More information

Convex Sets and Functions

Convex Sets and Functions B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line

More information

Riemann Integrals and the Fundamental Theorem of Calculus

Riemann Integrals and the Fundamental Theorem of Calculus Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums

More information

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

More information

Integrals along Curves.

Integrals along Curves. Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the

More information

Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integral Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

More information

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a). The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

More information

Mathematical Analysis: Supplementary notes I

Mathematical Analysis: Supplementary notes I Mthemticl Anlysis: Supplementry notes I 0 FIELDS The rel numbers, R, form field This mens tht we hve set, here R, nd two binry opertions ddition, + : R R R, nd multipliction, : R R R, for which the xioms

More information

NOTES AND PROBLEMS: INTEGRATION THEORY

NOTES AND PROBLEMS: INTEGRATION THEORY NOTES AND PROBLEMS: INTEGRATION THEORY SAMEER CHAVAN Abstrct. These re the lecture notes prepred for prticipnts of AFS-I to be conducted t Kumun University, Almor from 1st to 27th December, 2014. Contents

More information

Functions of bounded variation

Functions of bounded variation Division for Mthemtics Mrtin Lind Functions of bounded vrition Mthemtics C-level thesis Dte: 2006-01-30 Supervisor: Viktor Kold Exminer: Thoms Mrtinsson Krlstds universitet 651 88 Krlstd Tfn 054-700 10

More information

5.5 The Substitution Rule

5.5 The Substitution Rule 5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

More information

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230 Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b

x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick

More information

38 Riemann sums and existence of the definite integral.

38 Riemann sums and existence of the definite integral. 38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

More information

Chapter Gauss Quadrature Rule of Integration

Chapter Gauss Quadrature Rule of Integration Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to

More information

ON THE WEIGHTED OSTROWSKI INEQUALITY

ON THE WEIGHTED OSTROWSKI INEQUALITY ON THE WEIGHTED OSTROWSKI INEQUALITY N.S. BARNETT AND S.S. DRAGOMIR School of Computer Science nd Mthemtics Victori University, PO Bo 14428 Melbourne City, VIC 8001, Austrli. EMil: {neil.brnett, sever.drgomir}@vu.edu.u

More information

MA Handout 2: Notation and Background Concepts from Analysis

MA Handout 2: Notation and Background Concepts from Analysis MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,

More information

INTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).

INTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q). INTEGRALS JOHN QUIGG Eercise. Le f : [, b] R be bounded, nd le P nd Q be priions of [, b]. Prove h if P Q hen U(P ) U(Q) nd L(P ) L(Q). Soluion: Le P = {,..., n }. Since Q is obined from P by dding finiely

More information

Lecture 3. Limits of Functions and Continuity

Lecture 3. Limits of Functions and Continuity Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live

More information

Section 6.1 INTRO to LAPLACE TRANSFORMS

Section 6.1 INTRO to LAPLACE TRANSFORMS Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

More information

2 Definitions and Basic Properties of Extended Riemann Stieltjes Integrals

2 Definitions and Basic Properties of Extended Riemann Stieltjes Integrals 2 Definitions nd Bsic Properties of Extended Riemnn Stieltjes Integrls 2.1 Regulted nd Intervl Functions Regulted functions Let X be Bnch spce, nd let J be nonempty intervl in R, which my be bounded or

More information

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0. STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

More information

Introduction to Real Analysis (Math 315) Martin Bohner

Introduction to Real Analysis (Math 315) Martin Bohner ntroduction to Rel Anlysis (Mth 315) Spring 2005 Lecture Notes Mrtin Bohner Author ddress: Version from April 20, 2005 Deprtment of Mthemtics nd Sttistics, University of Missouri Roll, Roll, Missouri 65409-0020

More information

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri

More information

Necessary and Sufficient Conditions for Differentiating Under the Integral Sign

Necessary and Sufficient Conditions for Differentiating Under the Integral Sign Necessry nd Sufficient Conditions for Differentiting Under the Integrl Sign Erik Tlvil 1. INTRODUCTION. When we hve n integrl tht depends on prmeter, sy F(x f (x, y dy, it is often importnt to know when

More information

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests. ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

More information

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [, b] be bounded intervl in R. C 2 ([, b]) denotes the spce of functions with derivtives of second order continuous up to the endpoints. Cc 2

More information

On the Generalized Weighted Quasi-Arithmetic Integral Mean 1

On the Generalized Weighted Quasi-Arithmetic Integral Mean 1 Int. Journl of Mth. Anlysis, Vol. 7, 2013, no. 41, 2039-2048 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2013.3499 On the Generlized Weighted Qusi-Arithmetic Integrl Men 1 Hui Sun School

More information

Math 4200: Homework Problems

Math 4200: Homework Problems Mth 4200: Homework Problems Gregor Kovčič 1. Prove the following properties of the binomil coefficients ( n ) ( n ) (i) 1 + + + + 1 2 ( n ) (ii) 1 ( n ) ( n ) + 2 + 3 + + n 2 3 ( ) n ( n + = 2 n 1 n) n,

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

Big idea in Calculus: approximation

Big idea in Calculus: approximation Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:

More information

440-2 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam

440-2 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam 440-2 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP

More information

Math RE - Calculus II Area Page 1 of 12

Math RE - Calculus II Area Page 1 of 12 Mth --RE - Clculus II re Pge of re nd the Riemnn Sum Let f) be continuous function nd = f) f) > on closed intervl,b] s shown on the grph. The Riemnn Sum theor shows tht the re of R the region R hs re=

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.

More information

Lecture 14: Quadrature

Lecture 14: Quadrature Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

More information

A basic logarithmic inequality, and the logarithmic mean

A basic logarithmic inequality, and the logarithmic mean Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, Bbeş-Bolyi University Str. Koglnicenu

More information

Math Calculus with Analytic Geometry II

Math Calculus with Analytic Geometry II orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

More information

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by. NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

More information

Mapping the delta function and other Radon measures

Mapping the delta function and other Radon measures Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

More information

Test 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).

Test 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher). Test 3 Review Jiwen He Test 3 Test 3: Dec. 4-6 in CASA Mteril - Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 14-17 in CASA You Might Be Interested

More information

FINALTERM EXAMINATION 2011 Calculus &. Analytical Geometry-I

FINALTERM EXAMINATION 2011 Calculus &. Analytical Geometry-I FINALTERM EXAMINATION 011 Clculus &. Anlyticl Geometry-I Question No: 1 { Mrks: 1 ) - Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...

More information

Anonymous Math 361: Homework 5. x i = 1 (1 u i )

Anonymous Math 361: Homework 5. x i = 1 (1 u i ) Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define

More information

C1M14. Integrals as Area Accumulators

C1M14. Integrals as Area Accumulators CM Integrls s Are Accumultors Most tetbooks do good job of developing the integrl nd this is not the plce to provide tht development. We will show how Mple presents Riemnn Sums nd the ccompnying digrms

More information

u(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.

u(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C. Lecture 4 Complex Integrtion MATH-GA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex

More information

A Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions

A Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions Interntionl Journl of Mthemticl Anlysis Vol. 8, 2014, no. 50, 2451-2460 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2014.49294 A Convergence Theorem for the Improper Riemnn Integrl of Bnch

More information

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s). Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different

More information

QUADRATIC EQUATION. Contents

QUADRATIC EQUATION. Contents QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

More information

a n+2 a n+1 M n a 2 a 1. (2)

a n+2 a n+1 M n a 2 a 1. (2) Rel Anlysis Fll 004 Tke Home Finl Key 1. Suppose tht f is uniformly continuous on set S R nd {x n } is Cuchy sequence in S. Prove tht {f(x n )} is Cuchy sequence. (f is not ssumed to be continuous outside

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

RAM RAJYA MORE, SIWAN. XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII (PQRS) INDEFINITE INTERATION & Their Properties

RAM RAJYA MORE, SIWAN. XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII (PQRS) INDEFINITE INTERATION & Their Properties M.Sc. (Mths), B.Ed, M.Phil (Mths) MATHEMATICS Mob. : 947084408 9546359990 M.Sc. (Mths), B.Ed, M.Phil (Mths) RAM RAJYA MORE, SIWAN XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII

More information

UniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that

UniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de

More information

Henstock Kurzweil delta and nabla integrals

Henstock Kurzweil delta and nabla integrals Henstock Kurzweil delt nd nbl integrls Alln Peterson nd Bevn Thompson Deprtment of Mthemtics nd Sttistics, University of Nebrsk-Lincoln Lincoln, NE 68588-0323 peterso@mth.unl.edu Mthemtics, SPS, The University

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

A Mathematical Model for Unemployment-Taking an Action without Delay

A Mathematical Model for Unemployment-Taking an Action without Delay Advnes in Dynmil Systems nd Applitions. ISSN 973-53 Volume Number (7) pp. -8 Reserh Indi Publitions http://www.ripublition.om A Mthemtil Model for Unemployment-Tking n Ation without Dely Gulbnu Pthn Diretorte

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1 Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

More information

Calculus MATH 172-Fall 2017 Lecture Notes

Calculus MATH 172-Fall 2017 Lecture Notes Clculus MATH 172-Fll 2017 Lecture Notes These notes re concise summry of wht hs been covered so fr during the lectures. All the definitions must be memorized nd understood. Sttements of importnt theorems

More information