Familiarizing Students with Definition of Lebesgue Integral: Examples of Calculation Directly from Its Definition Using Mathematica

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1 Math.Comput.Sci. (27 : DOI.7/s Mathematics i Computer Sciece Familiarizig Studets with Defiitio of Lebesgue Itegral: Examples of Calculatio Directly from Its Defiitio Usig Mathematica Włodzimierz Wojas Ja Krupa Received: 7 December 26 / Revised: 9 April 27 / Accepted: 2 April 27 / Published olie: 3 May 27 The Author(s 27. This article is a ope access publicatio Abstract We preset i this paper several examples of Lebesgue itegral calculated directly from its defiitios usig Mathematica. Calculatio of Riema itegrals directly from its defiitios for some elemetary fuctios is stadard i higher mathematics educatio. But it is difficult to fid aalogical examples for Lebesgue itegral i the available literature. The paper cotais Mathematica codes which we prepared to calculate symbolically Lebesgue sums ad limits of sums. We also visualize the graphs of simple fuctios used for approximatio of the itegrals. We also show how to calculate the eeded sums ad limits by had (without CAS. We compare our calculatios i Mathematica with calculatios i some other CAS programs such as wxmaxima, MuPAD ad Sage for the same itegrals. Keywords Higher educatio Lebesgue itegral Applicatio of CAS Mathematica Mathematical didactics Mathematics Subject Classificatio 97R2 97I5 97B4 Itroductio Youg ma, i mathematics you do t uderstad thigs. You just get used to them Joh vo Neuma I hear ad I forget. I see ad I remember. I do ad I uderstad Chiese Quote Electroic Supplemetary material The olie versio of this article (doi:.7/s cotais supplemetary material, which is available to authorized users. W. Wojas J. Krupa (B Departmet of Applied Mathematics, Warsaw Uiversity of Life Scieces (SGGW, ul. Nowoursyowsa 59, Warsaw, Polad ja_rupa@sggw.pl W. Wojas wlodzimierz_wojas@sggw.pl

2 364 W. Wojas, J. Krupa I popular boos of calculus, for example [3,7,2], we ca fid may examples of Riema itegral calculated directly from its defiitio. The aim of these examples is to familiarize studets with the defiitio of Riema itegral. But we caot fid aalogical examples for Lebesgue itegral. I this article, with similar aim but for Lebesgue itegral defiitio, we preset the followig examples of calculatio directly from its defiitio: π/2 si x dm(x, exp(x dm(x, π l( 2r cos x +r 2 dm(x (r >, x dm(x ( N, x2 dm(x where dm(x deotes the Lebesgue measure o the real lie. The way we preseted we may also calculate e.g. x dm(x, x2 dm(x, x3 dm(x. We also cosider the calculatio of itegrals x2 dm(x, x/m dm(x (m N, b a χ [a,b]\q dm(x, e l x dm(x usig partitio the rage of f Lebesgue philosophy. We calculate sums, limits ad plot graphs of eeded simple fuctios usig Mathematica. 2 Defiitios of Lebesgue Itegral The followig two defiitios of Lebesgue itegral are used i the first part of this article: Let (R, M, m be the measure space, where M is σ -algebra of Lebesgue measurable subsets i R, ad m is the Lebesgue measure o R. Let R be exteded real umbers R with two more elemets adjoied, deoted + ad.leta be Lebesgue measurable subset of R. A real-valued fuctio f o A is called simple if it assumes oly fiitely may distict values. Let f : A R be measurable oegative fuctio (we ve omitted the defiitio of Lebesgue itegral for simple real measurable fuctios. Defiitio (See [4,6,8,,3] { f dm(x = sup s dm(x : s f, s : A R is simple measurable fuctio. (2. A A Defiitio 2 (See [,,5] Let s : A R be odecreasig sequece of oegative simple measurable fuctios such that lim s (x = f (x for every x A. The: f dm(x = lim s dm(x. (2.2 A A We stress the fact that the value of the Lebesgue itegral of fuctio f i the Defiitio 2 is idepedet of the choice of the sequece s. The proof of this fact ca be foud i [, p. 3, Theorem 7.5.] ad i [, p. 3], [5, p. 289, Theorem 4.6]. The equivalece of the two defiitios follows from the Lebesgue s Mootoe covergece theorem (See [3, Theorem.28, p. 38], [6] or it could be proved more elemetary (setch of the proof: From the basic properties of Lebesgue itegral for simple measurable fuctio we have that A s dm(x is odecreasig sequece of real umbers. Hece the limit lim A s dm(x exists (fite or. Suppose that lim A s dm(x = a for some odecreasig sequece s of oegative simple measurable fuctios such that lim s (x = f (x for every x A. Directly from properties of the least upper boud we have that { sup s dm(x : s f, s : A R is simple measurable fuctio A a. { Whe a = the the equivalece is obvious. Suppose that a < ad that sup A s dm(x : s f, s : A R is simple measurable fuctio > a. That meas that there exists simple measurable fuctio s such that s f ad A s(x dm(x >a.lett (x = max{s(x, s (x for all x A ad for =, 2,...

3 Familiarizig Studets with Defiitio of Lebesgue Itegral It ca be show that t are simple measurable fuctios for =, 2,..., s(x t (x t + (x for all x A, N ad lim t (x = f (x for all x A. Because the limit i the Defiitio 2 is idepedet of the choice of the sequece s ad a < A s(x dm(x A t (x dm(x we have: a < s(x dm(x lim t (x dm(x = lim s (x dm(x A A A which cotradicts the assumptio lim A s dm(x = a. it must be: { sup s dm(x : s f, s is simple measurable fuctio A = lim s (x dm(x. A We will cosider six examples of calculatig Lebesgue itegral directly from its defiitio. 3 Example: π/2 si x dm(x Let us cosider the fuctio: f (x = si x, x [,π/2. For the rest of this example we will restrict our cosideratio to x [,π/2. We will calculate π/2 si x dm(x applyig directly Defiitio. Cosider s (x = 2 = 2 ad s (x = si ( si 2 + π χ [ + 2+ π, (x, for x [,π/2, =, 2, π ( 2 + π χ [ 2 + π, (x, for x [,π/2, =, 2, π Usig Wolfram Mathematica we get the followig Figs. ad 2: Fig. Graphs of fuctios f, s, s 2. We ca see that s (x s 2 (x for x [,π/2

4 366 W. Wojas, J. Krupa Fig. 2 Graphs of fuctios f, s 2, s 3. We ca see that s 2 (x s 3 (x for x [,π/2 It is clear that s, s are sequeces of oegative simple measurable fuctios ad that s f ad s f o [,π/2 for all =, 2,... Usig Wolfram Mathematica we get: Listig Mathematica code: [ π I[]:= Simplify ( Out[]= 2 2 π = + Cot I[2]:= Limit[%, ] Out[2]= [ π ]] Si 2 + [ 2 2 π ] a = π/2 s dm(x = Similarly Listig 2 Mathematica code: [ π I[3]:= Simplify 2 = ( Out[3]= 2 2 π + Cot I[4]:= Limit[%, ] Out[4]= si π 2 + [ π ]] Si 2 + [ 2 2 π ] 2 + π = 2 2 π( + cot(2 2 π (3.

5 Familiarizig Studets with Defiitio of Lebesgue Itegral ā = π/2 2 s dm(x = si π 2 + ( 2 + π = 2 2 π + cot(2 2 π (3.2 Of course we could use the followig formulae: + si 2 si(x = x si 2 x si x si x ad lim x x = istead of 2 the code i Listigs ad 2 to get the results i formulae (3. ad (3.2. Usig formulae (3. ad (3.2, basic properties of least upper, greatest lower bouds ad properties of Lebesgue itegral of simple measurable fuctios we will prove that: { π/2 sup ad { π/2 sup s dm(x : s f, s is simple measurable fuctio (3.3 s dm(x : s f, s is simple measurable fuctio. (3.4 Iequalities (3.3 ad (3.4 give { π/2 sup s dm(x : s f, s is simple measurable fuctio =, which meas that π/2 f dm(x = π/2 { si x dm(x =. π/2 Because lim a =, so sup s dm(x : =, 2,... = sup a. { { π/2 π/2 From s dm(x : =, 2,... s dm(x : s f, s is simple measurable fuctio follows { π/2 sup s dm(x : s f, s is simple measurable fuctio. (3.5 For every simple measurable fuctio s such that s f we have also that s s, =, 2,... Hece for every simple measurable fuctio s such that s f we have π/2 s dm(x s dm(x, =, 2,..., hece { π/2 π/2 sup s dm(x : s f, s is simple measurable fuctio s dm(x, =, 2,..., hece { π/2 sup s dm(x : s f, s is simple measurable fuctio { π/2 if s dm(x, =, 2,.... (3.6 { π/2 The last iequality follows because lim ā =, which meas if s dm(x, =, 2,... = if ā. Iequalities (3.5 i(3.6 give { π/2 sup s dm(x : s f, s is simple measurable fuctio =, ad that meas π/2 f dm(x = π/2 si x dm(x =. Let calculate π/2 si x dm(x applyig directly Defiitio 2.

6 368 W. Wojas, J. Krupa We ca see that s (x s + (x for x [,π/2 ad for all =, 2,...IFigs. ad 2 we ca see that s (x s 2 (x ad s 2 (x s 3 (x for x [,π/2. We ca also see that lim s (x = si(x for all x [,π/2. s is odecreasig sequece of oegative simple measurable fuctios ad s coverges poitwise to f. by formula (3. ad directly by Defiitio 2 we get π/2 si x dm(x = π/2 f dm(x = lim π/2 s dm(x = lim a =. I the followig Sects. 4 8 we omitted graphs of fuctio f, s (x, s (x because they are similar to the previous oes i Sect Example: exp x dm(x Let cosider the fuctio: f (x = exp x, x [,. For the rest of this example we will restrict our cosideratio to x [,. We will calculate exp x dm(x applyig directly Defiitio. Cosider 2 ( s (x = exp 2 χ [ 2, (x, for x [,, =, 2, = 2 ad s (x = exp ( 2 χ [ 2, (x, for x [,, =, 2,... 2 It is clear that s, s are sequeces of oegative simple measurable fuctios ad that s f ad s f o [, for all =, 2,... Usig Wolfram Mathematica we get: Listig 3 Mathematica code: [ 2 [ ]] I[5]:= Simplify 2 Exp 2 Out[5]= 2 ( + e + e 2 = I[6]:= Limit[%, ] Out[6]= + e a = s dm(x = 2 = exp 2 2 = 2 ( e/( + e 2 e (4.

7 Familiarizig Studets with Defiitio of Lebesgue Itegral Similarly Listig 4 Mathematica code: [ 2 [ ]] I[7]:= Simplify 2 Exp 2 Out[7]= 2 ( + ee 2 + e 2 I[8]:= Limit[%, ] Out[8]= + e ā = 2 s dm(x = exp 2 2 = 2 e 2 ( e/( + e 2 e (4.2 Of course we could use the followig formulae: = q = q+ exp x q (q = ad lim x x = istead of the code i Listigs 3 ad 4 to get the results i formulae (4. ad (4.2. Usig formulae (4. ad (4.2 ad similar reasoig lie i the previous example (Sect. 3 we get: f dm(x = exp x dm(x = e directly from Defiitio ad exp x dm(x = f dm(x = lim s dm(x = lim a = e directly from Defiitio 2. 5 Example: π l( 2r cos x + r2 dm(x (r > Let cosider the fuctio: f (x = l( 2r cos x + r 2, x [,π, r >. For the rest of this example we will restrict our cosideratio to x [,π. We will calculate π l( 2r cos x + r 2 dm(x applyig directly Defiitio. Cosider s (x = ad s (x = 2 = 2 ( f 2 π χ [ 2 π, + 2 π (x, for x [,π, =, 2,... ( f 2 π χ [ 2 π, 2 π (x, for x [,π, =, 2,... It is clear that s, s are sequeces of oegative simple measurable fuctios ad that s f ad s f o [,πfor all =, 2,... I the Mathematica code below we use the fact that if s is a sequece of positive values with coverget sum, the we have s = l ( exp( s = l ( exp(s.

8 37 W. Wojas, J. Krupa Usig Wolfram Mathematica we get: Listig 5 Mathematica code: I[]:= g[x_] = 2 r Cos[x]+r 2 ; [ [ π ] /. { I[2]:= pr=simplify g 2 ] = ( + r ( + r 2+ Out[2]= + r I[3]:= d = r 2+ ; [ π [ pr ] ] [ π ] I[4]:= Limit 2 Log,, Assumptios r > +Limit d 2 Log[d],, Assumptios r > Out[4]= 2πLog[r] a = π Similarly s dm(x = 2 = Listig 6 Mathematica code: [ pr g[π] ( π f 2 + π 2π l(r (5. 2 ] I[5]:= pr=simplify g[] ( + r ( + r 2+ Out[5]= + r [ π [ pr ] ] [ π ] I[6]:= Limit 2 Log,, Assumptios r > +Limit d 2 Log[d],, Assumptios r > Out[6]= 2πLog[r] I listigs 5, 6 we used the substitutio rule (->2ˆ because whe we used directly 2ˆ istead, Mathematica could ot simplify the expressio. We caot calculate these limits i oe step usig Mathematica. But usig other CAS (wxmaxima, MuPAD we caot calculate these limits eve i two steps i ay way. ā = π s dm(x = 2 ( π f 2 + Of course we could use the followig formulae: z 2 = (z 2 ( 2z cos(π/ + z 2 π 2π l(r (5.2 2 istead of the code i Listigs 5 ad 6 to get the results i formulae (5. ad (5.2. Usig formulae (5. ad (5.2 ad similar reasoig lie i the previous example (Sect. 3 we get: π f dm(x = π l( 2r cos x + r 2 dm(x = 2π l(r directly from Defiitio ad π l( 2r cos x + r 2 dm(x = π π f dm(x = lim s dm(x = lim a = 2π l(r directly from Defiitio 2.

9 Familiarizig Studets with Defiitio of Lebesgue Itegral Example: xm dm(x (m N Let cosider the fuctio: f (x = x m, x [,, m N. For the rest of this example we will restrict our cosideratio to x [,. We will calculate xm dm(x applyig directly Defiitio. Cosider s (x = 2 = 2 ad s (x = ( m 2 χ [ 2, (x, for x [,, =, 2, ( 2 m χ [ 2, 2 (x, for x [,, =, 2,... As i the previous examples it is clear that s, s are sequeces of oegative simple measurable fuctios ad that s f ad s f o [, for all =, 2,... Usig Wolfram Mathematica we get: Listig 7 Mathematica code: [ 2 ( j m, ] I[]:= Limit, Assumptios m Itegers &&m > 2 2 Out[]= + m j= a = s dm(x = 2 = ( 2 m 2 m + (6. Similarly Listig 8 Mathematica code: [ 2 ( j m, ] I[2]:= Limit, Assumptios m Itegers &&m > 2 2 Out[2]= + m j= ā = 2 ( m s dm(x = 2 2 m + (6.2 Of course we could use the Stolz ad biomial theorems istead of the code i Listigs 7 ad 8 to get the results i formulae (6. ad (6.2. Usig formulae (6. ad (6.2 ad similar reasoig lie i the previous example (Sect. 3 we get: f dm(x = xm dm(x = directly from Defiitio m+ ad xm dm(x = f dm(x = lim s dm(x = lim a = m+ directly from Defiitio 2.

10 372 W. Wojas, J. Krupa 7 Example: x2 dm(x Let cosider the fuctio: f (x = x 2, x [,. For the rest of this example we will restrict our cosideratio to x [,. We will calculate x2 dm(x applyig directly Defiitio. Cosider s (x = 2 = 2 ( 2 ad s (x = 2 χ [ ( 2 2 χ [ [ + 2, 2 2, (x, for x [,, =, 2, [ 2, 2 2, 2 (x, for x [,, =, 2,... As i the previous examples it is clear that s, s are sequeces of oegative simple measurable fuctios ad that s f ad s f o [, for all =, 2,... Usig Wolfram Mathematica we get: Listig 9 Mathematica code: I[]:= Out[]= 3 ( ( I[2]:= Limit[%, ] Out[2]= 2 3 a = s dm(x = 2 = ( (7. Similarly Listig Mathematica code: I[3]:= Out[3]= ( + 2 ( I[4]:= Limit[%, ] Out[4]= 2 3 ā = 2 ( 2 2 s dm(x = (7.2

11 Familiarizig Studets with Defiitio of Lebesgue Itegral Of course we could use the formula 2 = 6( + (2 + istead of the code i Listigs 9 ad to get the results i formulae (7. ad (7.2. Usig formulae (7. ad (7.2 ad similar reasoig lie i the previous example (Sect. 3 we get: f dm(x = x2 dm(x = 2 3 directly from Defiitio ad x2 dm(x = f dm(x = lim s dm(x = lim a = 2 3 directly from Defiitio 2. 8 Example: f (x dm(x, where f (x is Thomae s fuctio Let us cosider Thomae s fuctio if x =, f (x = q if x = p q [, ] Q, p q if x [, ]\Q. is i lowest terms, (8. The Thomae s fuctio f (x is Riema itegrable over [, ] ad R f (x dx =. Let us calculate f (x dm(x directly from Defiitio. Let < 2 < 2 2 < < 2 2 < be partitio of [, ]. LetA ={x [, ] : 2 f (x < 2 = f ([ 2, 2 for =, 2,...,2 ad A 2 ={x [, ] : 2 2 f (x = f ([ 2 2, ]. Oe ca see that A =[, ]\B, where B Q ad A [, ] Q for = 2, 3,...,2. Hece m(a =, m(a = for = 2, 3,...,2, where m is the Lebesgue measure o R. Cosider s (x = ad s (x = χ A (x, for x [,, =, 2,... 2 χ A (x, for x [,, =, 2,... As i the previous examples it is clear that s, s are sequeces of oegative simple measurable fuctios ad that s f ad s f o [, for all =, 2,... Oe ca see that: a = ad ā = s dm(x = s dm(x = m(a =. (8.2 2 m(a =. (8.3 2 Usig formulae (8.2 ad (8.3 ad similar reasoig lie i the previous example (Sect. 3 we get: f dm(x = directly from Defiitio ad f dm(x = lim s dm(x = directly from Defiitio 2.

12 374 W. Wojas, J. Krupa 9 The Lebesgue Philosophy The Lebesgue philosophy was well described i literature e.g. we will follow the descriptio i [5, Chapter 3]. Lebesgue s goal was to collect approximately equal values of f. He proceeded i the followig way. Let f : [a, b] R be a bouded Lebesgue measurable fuctio, ad cosider the partitio Q : c = y < y < < y = d where c = if{ f (x : x [a, b] ad d = sup{ f (x : x [a, b] ad the diameter of Q is Q =max{ y y : =,...,. If A ={x [a, b] :y f (x <y, =,..., ad A ={x [a, b] :y f (x d we defie s Q = y m(a ad S Q = y m(a, where m is the Lebesgue measure o R ad L b a b f dm(x = sup s Q ad L f dm(x = if S Q. Q a Q Defiitio 3 A bouded Lebesgue measurable fuctio f :[a; b] R is Lebesgue itegrable if L b f dm(x = L b a f dm(x. I this case, the Lebesgue itegral of f over [a; b] is b b L f dm(x = L f dm(x = L a a b a f dm(x. a 9. The Lebesgue Philosophy for x2 dm(x Let us calculate x2 dm(x directly from Defiitio 3. Let f (x = x 2 ad g(x = f (x = x for x [, ]. Let y = /, =,, 2,..., so Q : c = < < < < = d. s Q = y m(a = ( y g(y g(y = = (g(y g(y = ( / g(y = ( g( g(/ / = ( g(y g(y g(/. (9.

13 Familiarizig Studets with Defiitio of Lebesgue Itegral Usig Stolz theorem we calculate the limit: lim 3/2 = lim = lim ( + + ( + 3/2 + 3/2 ( + 3/2 = lim 3/2 ( + 3 ( 3 + ( + 3/2 + 3/2 ( + / ( + / 3/ = lim / + / 2 = 2/3. (9.2 lim s Q have = 2/3 = /3. Hece from the basic properties of s Q, S Q ad least upper boud we sup s Q /3 ad S Q /3 for all partitios Q. (9.3 Q Similarly S Q = lim y m(a = ( y g(y g(y = = = ( g(y g(y g(y = ( g( g( = ( g(y g(y = g(/ = ( / =. (9.4 3/2 Usig Stolz theorem we calculate the limit: 3/2 = lim = lim (( + 3/2 + 3/2 ( + 3/2 = lim 3/2 ( ( ( + 3/2 + 3/2 3 2 = lim lim S Q = 2/3 = /3. Hece similarly we have ( + / 3/ / + /2 = 2/3. (9.5 if Q S Q /3 ad s Q /3 for all partitios Q. (9.6 From 9.3 ad 9.6 follows that: L x 2 dm(x = sup s Q = /3 ad L Q Hece directly from Defiitio 3 we have L x 2 dm(x = 3. x 2 dm(x = if Q S Q = /3. We caot get Wolfram Mathematica, wxmaxima ad MuPAD to calculate the above limits. I Figs. 3 ad 4 below we preset dyamic plots of Lebesgue lower ad upper sums s Q ad S Q for L x2 dm(x created usig Wolfram Mathematica. We used Maipulate fuctio. The above calculatio are much more difficult that the oes for Riema itegral x2 dx. Similarly we ca calculate x3 dm(x ad xm dm(x (m N.

14 376 W. Wojas, J. Krupa Fig. 3 Dyamic plots of Lebesgue lower sums s Q for L x2 dm(x ad {2, 2, 4, 6,, 4, Fig. 4 Dyamic plots of Lebesgue upper sums S Q for L x2 dm(x ad {2, 2, 4, 6,, 4,

15 Familiarizig Studets with Defiitio of Lebesgue Itegral The dyamic versios of the Figs. 3 ad 4 ca be foud i Electroic supplemetary material. Actually whe we tae the followig rage partitio y = ( 2, =,,...,the calculatio for the cosidered Lebesgue itegral x2 dm(x are pretty the same lie for the Riema oe (slightly differet reasoig. Similarly whe we tae the followig rage partitio y = si ( 2 π, =,,..., the calculatio for the cosidered Lebesgue itegral π/2 si x dm(x are pretty the same lie for the Riema case. We ca see graphical presetatio of the Lebesgue philosophy doe i [2,9]. But we could ot fid examples calculated directly from Defiitio 3 except for the Dirichlet lie fuctio. 9.2 The Lebesgue Philosophy for b a χ [a,b] Q dm(x I [5] is proved that Dirichlet fuctio χ [a,b] Q is ot Riema itegrable but it is proved directly from Defiitio 3 that L b a χ [a,b] Q dm(x =. 9.3 The Lebesgue Philosophy for b a χ [a,b]\q dm(x Let f (x = χ [a,b]\q (x = { if x [a, b]\q, if x [a, b] Q. Let us calculate b a χ [a,b]\q dm(x directly from Defiitio 3. Let Q : = y < y < < y = be ay partitio of [, ]. Oe ca see that A =[a, b] Q, A 2 = A 3 = = A =, A =[a, b]\q. Thus s Q = y (b a, S Q = (b a for ay partitio Q. sup Q s Q = b a ad if Q S Q = b a. Hece L b a χ [a,b]\q dm(x = b a directly from Defiitio The Lebesgue Philosophy for f (x dm(x, where f (x is Thomae s fuctio Let us cosider Thomae s fuctio f (x which is defied i formula 8. i Sect. 8. Let us calculate f (x dm(x directly from Defiitio 3. Let Q : = y < y < < y = be ay partitio of [, ]. Oe ca see that A =[, ]\B, where B Q ad A 2 [, ] Q, A 3 [, ] Q,..., A [, ] Q. Hece m(a =, m(a 2 = m(a 3 = =m(a =, where m is the Lebesgue measure o R. Thus s Q =, S Q = y for ay partitio Q. sup Q s Q = ad if Q S Q =. Hece L f (x dm(x = directly from Defiitio The Lebesgue Philosophy for x/m dm(x; (m N Let us calculate x/m dm(x; m N directly from Defiitio 3.

16 378 W. Wojas, J. Krupa Let f (x = x /m ad g(x = f (x = x m for x [, ]. Let y = /, =,, 2,..., so Q : c = < < < < = d. s Q = y m(a = ( y g(y g(y = = (g(y g(y = Similarly S Q = g(y = ( g( g(/ ( m (/ m = m+ y m(a = ( y g(y g(y = = m+ = ( g(y g(y g(y = ( g( g( = ( m (/ m = = Usig Wolfram Mathematica we get: Listig Mathematica code: [ I[]:= Limit Out[]= + m m+ ( g(y g(y g(/ m. (9.7 ( g(y g(y = g(/ m. (9.8 ( m, ], Assumptios m Itegers &&m > ad Listig 2 Mathematica code: [ I[2]:= Limit Out[2]= + m ( m, ], Assumptios m Itegers &&m > From listigs, 2 ad we have lim s Q = m + = m m + ad lim S Q = m + = m m + (9.9 Of course we could use the Stolz ad biomial theorems istead of the code i Listigs ad 2 to get the results i formulae (9.9.

17 Familiarizig Studets with Defiitio of Lebesgue Itegral Similar reasoig lie i the previous example gives: L x /m dm(x = sup s Q = m ad Q m + Hece directly from Defiitio 3 we have L x /m dm(x = m m +. L x /m dm(x = if S Q = m Q m The Lebesgue Philosophy for e l x dm(x Let us calculate e l x dm(x directly from Defiitio 3. Let f (x = l x x [, e] ad g(x = f (x = exp x for x [, ]. Let y = /, =,, 2,..., so Q : c = < < < < = d. s Q = y m(a = Similarly S Q = y m(a = ( y g(y g(y = ( y g(y g(y = Usig Wolfram Mathematica we get: Listig 3 Mathematica code: [ I[]:= Limit Out[]= ( [ ] [ ] ] ( Exp Exp, ( exp(/ exp ( ( /. (9. ( exp(/ exp ( ( /. (9. ad Listig 4 Mathematica code: [ I[2]:= Limit Out[2]= ( [ ] Exp Exp [ ] ], Of course we could use the formula for sum of geometric series istead of the code i Listigs 3 ad 4. From listigs 3, 4 ad similar reasoig lie i the last sectio we have e L l x dm(x = sup s Q = ad L Q Hece directly from Defiitio 3 we have L e l x dm(x =. e l x dm(x = if Q S Q =.

18 38 W. Wojas, J. Krupa Comparig the Calculatios i Mathematica with Calculatios i me Other CAS Programs The authors tried to repeat calculatios for examples of Lebesgue itegrals usig some other CAS programs such as: wxmaxima, MuPAD ad Sage. The followig Lebesgue itegrals: π/2 si x dm(x, ex dm(x, x2 dm(x were calculated i aalogical way i these CAS programs. We used stadard procedures i these programs to calculate sums ad limits. We could ot calculate the itegrals: xm dm(x (m N ad π l( 2r cos x + r 2 dm(x (r >. The use of procedures which we have used before (for itegrals π/2 si x dm(x, ex dm(x, x2 dm(x for these two itegrals had o effect. We used stadard procedures: sum, limit, simplificatio, assume. Coclusios I this paper the authors preset several examples of Lebesgue itegral calculated directly from its defiitio usig Mathematica. We also cosider the calculatio of itegrals usig partitio the rage of f Lebesgue-philosophy. Familiarizig studets with defiitio of a itegral by calculatio itegrals directly from its defiitio is a stadard approach i the case of Riema itegral. May examples of Riema itegral calculated usig oly its defiitio ca be foud i literature. We could ot fid ay aalogical examples i available literature for Lebesgue itegral, so this paper is a attempt to fill this gap. Usig Mathematica or other CAS programs for calculatio Lebesgue itegrals directly from its defiitios, seems to be didactically useful for studets because of the possibility of symbolic calculatio of sums, limits ad plot graphs checig our had calculatios. Moreover we get studets used ot oly to defiitio of Lebesgue itegral but also to CAS applicatios geerally. The priciples of programmig i Mathematica laguage are give i [4,6]. The six examples from Sects. 3 8 of our article could be used as a supplemet to the textboos [4,6,8,,3] (whe we use Defiitio ad also as a supplemet to to the textboos [,,5] (whe we use Defiitio 2. Ad this is the mai reaso why we i our article icluded the Defiitios ad 2 of Lebesgue itegral ad calculated the examples of itegral directly from these defiitios. Acowledgemets The authors would lie to express sicere gratitude to their referees for may valuable suggestios ad helpful criticism of the earlier versio of this wor. Ope Access This article is distributed uder the terms of the Creative Commos Attributio 4. Iteratioal Licese ( creativecommos.org/liceses/by/4./, which permits urestricted use, distributio, ad reproductio i ay medium, provided you give appropriate credit to the origial author(s ad the source, provide a li to the Creative Commos licese, ad idicate if chages were made. Refereces. Alipratis, C.D., Burishaw, O.: Priciples of Real Aalysis, 2d ed. Academic Press, Cambridge (99 2. Atoov, A.: Mathematica presetatio of Lebesgue itegratio Apostol, T.M.: Calculus, Volume, Oe-Variable Calculus with a Itroductio to Liear Algebra, 2d ed. Addiso-Wesley Publishig Compay, Bosto (99 4. Bartle, R.G.: The Elemets of Itegratio ad Lebesgue Measure. Wiley, Hoboe ( Beedetto, J.J., Czaja, W.: Itegratio ad Moder Aalysis. Birhäuser, Bosto, MA (29 6. Browder, A.: Mathematical Aalysis a Itroductio, 2d ed. Spriger, Berli (2 7. Fichteholz, G.M.: Differetial ad Itegral Calculus, 3rd ed. Fizmatgiz, Moscow ( Follad, G.B.: Real Aalysis Moder Techique, 2d ed. Wiley, Hoboe (27 9. Freiche, F.J.: Mathematica demostratio of Riema versus Lebesgue itegratio. RiemaVersusLebesgue/. Joes, F.: Lebesgue Itegratio o Euclidea Space. Joes & Bartlett Learig, Burligto (2. Kołodziej, W.: Mathematical Aalysis. Polish Scietific Publishers PWN, Warsaw (22. (i Polish 2. Larso, R.E., Hostetler, R.P., Edwards, B.H.: Calculus, 6th ed. Houghto Miffli Compay, Bosto (998

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