CAVALIERI INTEGRATION

Transcription

1 CAVALIERI INTEGRATION T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER Abstrct. We use Cvlieri s principle to develop novel integrtion technique which we cll Cvlieri integrtion. Cvlieri integrls differ from Riemnn integrls in tht non-rectngulr integrtion strips re used. In this w we cn use single Cvlieri integrls to find the res of some interesting regions for which it is difficult to construct single Riemnn integrls. We lso present two methods of evluting Cvlieri integrl b first trnsforming it to either n equivlent Riemnn or Riemnn-Stieltjes integrl b using specil trnsformtion functions h() nd its inverse g(), respectivel. Interestingl enough it is often ver difficult to find the trnsformtion function h(), wheres it is ver simple to obtin its inverse g().. Introduction We will use Cvlieri s principle to develop novel integrtion technique which cn be used to lmost effortlessl find the re of some interesting regions for which it is rther difficult to construct single Riemnn integrls. We will cll this tpe of integrtion Cvlieri integrtion. As the nme suggests, Cvlieri integrtion is bsed on the well known Cvlieri principle, stted here without proof []: Theorem. (Cvlieri s principle). Suppose two regions in plne re included between two prllel lines in tht plne. If ever line prllel to these two lines intersects both regions in line segments of equl length, then the two regions hve equl res. A B Figure. Simple illustrtion of Cvlieri s principle in R, with re A re B. Inspired b Cvlieri s principle, we pose the following question: wht hppens when we replce the usul rectngulr integrtion strip of the Riemnn sum with n integrtion strip tht hs non-rectngulr shpe? It turns out tht such formultion leds to consistent scheme of integrtion with few surprising results. 99 Mthemtics Subject Clssifiction. Primr 6A. Ke words nd phrses. Cvlieri; method of indivisibles; integrtion; Riemnn; Riemnn- Stieltjes.

2 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER B considering non-rectngulr integrtion strips we form Cvlieri sum which cn either be trnsformed to norml Riemnn sum (of n equivlent region) b using trnsformtion function h(), or to Riemnn-Stieltjes sum b using the inverse trnsformtion function g(). The min result of Cvlieri integrtion cn be demonstrted b using simple emple. Consider the region bounded b the -is nd the lines f(), nd b(), shown in Figure.A. Notice tht we cnnot epress the re of this region s single Riemnn integrl. We cn however clculte the re of this region b using single Cvlieri integrl: Are f() d, which is relted to Riemnn integrl nd Riemnn-Stieltjes integrl s follows: f() d f h() d f() dg(). For the present emple we hve the following result, since h() / nd g() : d d d The trnsformed regions f h() (corresponding to the Riemnn formultion) nd f() g () (corresponding to the Riemnn-Stieltjes formultion) re shown in Figure.B nd Figure.C, respectivel. f()d b() f() b A. f() h()d b B. f()g ()d b C. Figure. Illustrtion of Cvlieri integrtion b emple. In this pper we will show how to find the trnsformtion function h() nd its inverse g(). We first give brief overview of clssicl integrtion theor (Section ), followed b the derivtion of Cvlieri integrtion in Section. Finll we present number of full worked emples in Section, which clerl demonstrte how Cvlieri integrtion cn be pplied to vriet of regions.. Clssicl Integrtion Theor One of the oldest techniques for finding the re of region is the method of ehustion, ttributed to Antiphon []. The method of ehustion finds the re of region b inscribing inside it sequence of polgons whose res converge to the re of the region. Even though clssicl integrtion theor is well estblished

3 CAVALIERI INTEGRATION field there re still new results being dded in modern times. For emple, in the ver interesting pper b Ruff [5] the method of ehustion ws generlized, which led to n integrtion formul tht is vlid for ll Riemnn integrble functions: f()d (b ) n ( ) m+ n f n m ( + ) m(b ) n. Clssicl integrtion theor is however ver different from the method of ehustion, nd is minl ttributed to Newton, Leibniz nd Riemnn. Newton nd Leibniz discovered the fundmentl theorem of Clculus independentl nd developed the mthemticl nottion for clssicl integrtion theor. Riemnn formlized clssicl integrtion b introducing the concept of limits to the foundtions estblished b Newton nd Leibniz. However, the true fther of clssicl integrtion theor is probbl Bonventur Cvlieri (598 67). Cvlieri devised methods for computing res b mens of indivisibles []. In the method of indivisibles, region is divided into infinitel mn indivisibles, ech considered to be both one-dimensionl line segment, nd n infinitesimll thin two-dimensionl rectngle. The re of region is then found b summing together ll of the indivisibles in the region. However, Cvlieri s method of indivisibles ws hevil criticized due to the indivisible prdo, described net []... Indivisible prdo. Consider sclene tringle, ABC, shown in Figure.A. B dropping the ltitude to the bse of the tringle, ABC is prtitioned into two tringles of unequl re. If both the left ( ABD) nd right ( BDC) tringles re divided into indivisibles then we cn esil see tht ech indivisible (for emple EF ) in the left tringle corresponds to n equl indivisible (for emple GH) in the right tringle. This would seem to impl tht both tringles must hve equl re! A E F B D G H A. C A B D B. C A I B D J K C. C Figure. Cvlieri s indivisible prdo. Of course this rgument is clerl flwed. To see this, we cn investigte it more closel from mesure-theoretic point of view, s shown in Figure.B. Drwing strip of width through the tringle nd clculting the pre-imge of this strip produces two intervls on the -is with unequl width. Letting produces the two indivisibles EF nd GH. However, it does not mtter how smll ou mke, the two intervl lengths nd will never be equl. In other words, the re tht EF nd GH contributes to the totl re of the tringle must be different. There is n even simpler w to renounce the bove prdo: insted of using indivisibles prllel to the -is, we use indivisibles prllel to BC, s shown in Figure.C. Then ech pir of corresponding indivisibles IJ in ABD nd JK

4 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER in BDC clerl hs different lengths lmost everwhere. Therefore the res of ABD nd BDC need not be the sme. This trick of considering indivisibles (or infinitesimls) other thn those prllel to the -is forms the bsis of Cvlieri integrtion, in which non-rectngulr integrtion strips will be used.. Cvlieri Integrtion We present method of integrtion which we will refer to s Cvlieri integrtion, in which the primr difference from ordinr Riemnn integrtion is tht more generl integrtion strips cn be used. In some sense the Cvlieri integrl cn lso be seen s generliztion of the Riemnn integrl, in tht the Cvlieri formultion reduces to the ordinr Riemnn integrl when the integrtion strips re rectngulr. Tht is not to s tht the Cvlieri integrl etends the clss of Riemnn-integrble functions. In fct, the clss of Cvlieri-integrble functions is ectl equivlent to the clss of Riemnn-integrble functions. However, the Cvlieri integrl llows us to epress the res of some regions s single integrls for which we would hve to write down multiple ordinr Riemnn integrls... Preliminries nd Definitions. In order to develop (nd clerl present) the Cvlieri integrtion theor, number of definitions must first be introduced. Also note tht we will restrict our ttention to integrtion in R, with coordinte es nd. Definition. (Trnsltionl function). A continuous function is clled trnsltionl function with respect to continuous function f() on the intervl [, b] if { f() + z } is singulr, for ever z (b ) nd (). The bove definition ss tht n continuous function which intersects continuous function f() ectl once for n rbitrr trnsltion on the -is within the intervl [, b] is clled trnsltionl function. Two emples of trnsltionl functions re shown in Figure.A nd Figure.B, nd Figure.C presents n emple of liner function which is not trnsltionl with respect to f(). f() f() f() b b b A. B. C. Figure. Emples of trnsltionl, nd non-trnsltionl functions. Definition. (Cvlieri region R). Let R be n region (in R ) bounded b nonnegtive function f() (which is continuous on the intervl [, b ]), the -is, nd the boundr functions nd b(), where is trnsltionl function, b() : + (b ). Furthermore we hve tht nd b re the unique -vlues for which nd b() intersect f(), respectivel; nd () nd b b(). Then R is clled the Cvlieri region bounded b f(),, b() nd the -is.

5 CAVALIERI INTEGRATION 5 φ f E R b() b b Figure 5. A Cvlieri region R with integrtion boundries nd b(), nd n equivlent region E with integrtion boundries nd b. The Cvlieri integrl (which we will formll define in Definition.) cn be relted to n ordinr Riemnn integrl through prticulr trnsformtion h, which we will consider in some detil below. It m be useful to think of this trnsformtion (t lest intuitivel) s trnsforming n Cvlieri region R into n equivlent region E with equl re (see Figure 5), but with integrtion boundries nd b. Tht is, the re of the equivlent region E cn esil be epressed in terms of n ordinr definite integrl φ() d. Definition. (Trnsformtion function h). Let be trnsltionl function. The mpping h : [, b] [, b ], which mps i [, b] to i [, b ], is defined s h( i ) : { i [, b ] : (f i ) + [ i ] i, ()}, which we will refer to s the trnsformtion function (we will prove tht it is indeed function below). Proposition.. The mpping h : [, b] [, b ] is function. Proof. Tht h is function follows directl from the definition of trnsltionl function (Definition.), since we know tht { f( i ) + [ i ] i } must be singulr for ever [ i ] (b ). Tht is, h mps ever point i [, b] to ectl one point i [, b ]. Proposition.5. The trnsformtion function h is strictl monotone on [, b]. Proof. Let R be Cvlieri region bounded b f(),, b() nd the -is, s shown in Figure 6. Two possibilities m rise. Cse I: { f() } h is strictl incresing: Consider n trnsltion of, + c, s.t. + c (, b). Since the domin D(f), nd since intersects f() t, the trnsltion + c cnnot lso intersect f() t. Insted, we clerl hve tht + c must intersect f() t point c > on D(f). We now define A s the region bounded b the trnsltionl functions nd + c, nd the lines nd b (see Figure 6). The continuous function f() on the intervl [, b ] must lie within the region A, since n point of f() outside of this region would impl tht cnnot be trnsltionl function. Tht is, if f() hs points outside of region A, then there eists trnsltion of s.t. intersects f() t more thn one point.

6 6 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER Now consider n trnsltion of, + d, where d > c nd d (, b]. Suppose tht + d induces point d, with < d < c. Tht is, + d intersects f() t some point in region A. The functions + c nd + d re continuous on the intervl [, γ], where γ : { f() + d } (in fct, n trnsltionl function must be continuous on R). Now let Ψ : + c ( + d ), which is gin continuous function on [, γ]. Since c < d nd c > d b ssumption, it follows tht Ψ() < nd Ψ(γ) >. From the intermedite vlue theorem it follows tht there eists point α [, γ] s.t. Ψ(α). Tht is, (α) + c (α) + d. But this is impossible, since + d is trnsltion of + c. Therefore c h(c) < h(d) d. Since c is rbitrr nd d > c h(d) > h(c), h is strictl incresing on [, b]. Cse II: { f() } b h is strictl decresing: The second cse cn be proved in similr mnner s Cse I bove, in which cse h is strictl decresing function with the order of the induced prtition P reversed. Since h is either strictl incresing (Cse I) or strictl decresing (Cse II), it is strictl monotone on [, b]. γ A + c α f() + d b() d c c d b b c d Figure 6. Sketch for the proof of Proposition.5. Proposition.6. The trnsformtion function h is continuous on [, b]. Proof. Choose n rbitrr vlue [, b] such tht +c. We cn now define sequence ( i ) with i + i, i N. Now ( i ) s i. The sequence of functions ( + [ i + c] ) hs -intercepts equl to ( i ). The mpping h now genertes new sequence ( i ) s.t. i, i { i : f( i )+[ i +c] i }.

7 CAVALIERI INTEGRATION 7 Now tking the limit s i lim i i lim [ f( i ) + ( i + c) i ] i lim i [ f( i ) + ( i + c) i ] [ f( i ) + ( lim i i + c) i ] [ f( i ) + c) f( i )] [( i ) + [ ]) f( i )] This shows tht i s i ssuming [ f( i ) + [ ]) f( i )] hs one unique solution, which must be the cse since is trnsltionl function. The function h must be continuous t since i s i. Since is rbitrr, h is continuous function on [, b]. Proposition.7. The trnsformtion function h is bijective on [, b]. Proof. Tht h is injective on [, b] follows from the fct tht h is strictl monotone on [, b] (b Proposition.5). Furthermore h is clerl surjective on [, b], since it is continuous on [, b] (b Proposition.6). Since h is both injective nd surjective on [, b], h is lso bijective on [, b]... Derivtion of Cvlieri Integrtion. Since we wnt to derive the Cvlieri integrl which uses more generl integrtion strips thn the rectngles of the Riemnn integrl we first need to formll define vlid integrtion strips. Definition.8 (Integrtion strip). An integrtion strip is n re bounded below b the -is, on the left b trnsltionl function w.r.t. f() on [, b], from the right b b() + (b ), nd from bove b the line c. An emple of three integrtion strips is given in Figure 7, where Figure 7. corresponds to the usul Riemnn integrtion strip. c b() b() () (b) (c) b() Figure 7. Three integrtion strips with integrtion boundries nd b(). From Cvlieri s principle it follows tht we cn esil compute the re of n integrtion strips. Proposition.9 (Cvlieri s principle for integrtion strips.). The re of n integrtion strip is equl to A (b )c.

8 8 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER Proof. The re of n integrtion strip cn be determined b clculting the re between the curves b() nd with the definite integrl A c c b() d (b ) d (b ) (b )c. In order to find the re of Cvlieri region R, we need to ssocite two relted prtitions P nd P to the region R. Definition.. A prtition of [, b] is finite set P of points,,..., n such tht < < < n b. We describe P b writing: c P {,,..., n }. The n subintervls into which prtition P {,,..., n } divides [, b] re [, ], [, ],..., [ n, n ]. Their lengths re,,..., n n, respectivel. We denote the length k k of the kth subintervl b k. Thus nd we define k k k :. We now choose n prtition P {,,..., n} of [, b], nd we inscribe over ech subintervl derived from P the lrgest integrtion strip tht lies inside the Cvlieri region R. Since both boundries of n integrtion strip re necessril trnsltions of the trnsltionl function, we cn ppl the trnsformtion function h to the prtition P. If the trnsformtion function h is strictl incresing, the restriction of h to the prtition P induces new prtition P {,,..., n} s shown in Figure 8. Otherwise, if h is strictl decresing, the restriction of h induces reversed prtition P { n, n,..., }. In the rest of this document we will ssume tht h is strictl incresing, without n loss of generlit. n k k n f R R R R k m k R n k k k n b n Figure 8. The lower Cvlieri sum of f() for prtition P on [, b].

9 CAVALIERI INTEGRATION 9 Since we hve ssumed tht f is continuous nd nonnegtive on [, b ], we know from the Mimum-Minimum theorem tht for ech k between nd n there eists smllest vlue m k of f on the kth subintervl [ k, k ]. If we choose m k s the height of the kth integrtion strip R k, then R k will be the lrgest (tllest) integrtion strip tht cn be inscribed in R over [ k, k ]. Doing this for ech subintervl, we crete n inscribed strips R, R,..., R n, ll ling inside the region R. For ech k between nd n the strip R k hs bse [ k, k ] with width k nd hs height m k. Hence the re of R k is the product m k k (b Cvlieri s principle). The sum L(P, f, h) m k k, (lower Cvlieri sum) k where m k inf f(), h( i ) i i h( i ) is clled the lower Cvlieri sum nd should be no lrger thn the re of R. The lower Cvlieri sum is represented grphicll in Figure 8. Recll tht the lower Riemnn sum is defined similrl, tht is L(P, f) m k k, (lower Riemnn sum) k where m k inf f(), i i nd P {,,..., n } is prtition on [, b], nd the integrtion strips re rectngulr. The lower Riemnn sum is represented grphicll in Figure 9. f R R R R k m k R n k k k n n b Figure 9. The Lower Riemnn Sum of f() for prtition P on [, b]. Irrespective of how we define the re of the Cvlieri region R, this re must be t lest s lrge s the lower Cvlieri sum L(P, f, h) ssocited with n prtition P of [, b]. B procedure similr to the one tht involves inscribing integrtion strips to compute lower Cvlieri sum, we cn lso circumscribe integrtion strips nd compute n upper Cvlieri sum s shown in Figure. Let P {,,..., n} be given prtition of [, b], nd let f be continuous nd nonnegtive on [, b ]. The Mimum-Minimum Theorem implies tht for ech k between nd n there eists lrgest vlue M k of f on the kth integrtion strip

10 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER n k k n f R R R R k M k R n k k k n b n Figure. The upper Cvlieri sum of f() for prtition P on [, b]. R k, such tht R k will be the smllest possible strip circumscribing the pproprite portion of R. The re of R k is M k k, nd the sum U(P, f, h) M k k, (upper Cvlieri sum) k where M k sup f(), h( i ) i i h( i ) is clled the upper Cvlieri sum of f ssocited with the prtition P. The upper Cvlieri sum is represented grphicll in Figure. Recll tht the upper Riemnn sum is defined similrl, tht is U(P, f) M k k, (upper Riemnn sum) k where M k sup f(), i i nd P {,,..., n } is prtition on [, b], nd the integrtion strips re rectngulr. The upper Riemnn sum is represented grphicll in Figure. f R R R R k M k R n k k k n n b Figure. The upper Riemnn sum of f() for prtition P on [, b]. Irrespective of how we define the re of the Cvlieri region R, this re must be no lrger thn the upper Cvlieri sum U(P, f, h) for n prtition P of [, b].

11 CAVALIERI INTEGRATION The ssumption tht f must be nonnegtive on [, b ] cn now be dropped. Assuming onl tht f is continuous on [, b ], we still define the lower nd upper Cvlieri sums of f for prtition P of [, b] b L(P, f, h) m k k nd U(P, f, h) k M k k, where for n integer k between nd n, m k nd M k re the minimum nd mimum vlues of f on [ k, k ], respectivel. Remrk.. In the rest of this document we will repetedl mke use of the following nottion. We will let f() be n continuous function on the intervl [, b ]. We will lso ssume tht is some trnsltionl function w.r.t. f() on the intervl [, b], with which we ll ssocite prtition P. Furthermore, we will let h denote the trnsformtion function which mps the prtition P [, b] to the prtition P [, b ]. Of course, b() must be prticulr trnsltion on the -is of, such tht b() + (b ), where () nd b b() s defined previousl. Finll, we hve tht nd b re the unique -vlues for which nd b() intersect f(), respectivel. Definition. (Cvlieri sum). For ech k N from to n, let t k be n rbitrr number in [ k, k ] [, b ]. Then the sum C(P, f, h) k f(t k) k f(t ) + f(t ) + + f(t n) n k is clled Cvlieri sum for f on [, b]. Recll tht Riemnn sum for f on [, b] is defined similrl, tht is R(P, f) f(t k ) k f(t ) + f(t ) + + f(t n ) n, k where P {,,..., n } is n prtition of [, b], nd t k is n rbitrr number in [ k, k ] [, b]. Proposition.. The lower Cvlieri sum L(P, f, h) is equivlent to the lower Riemnn sum L(P, f h), tht is (.) L(P, f, h) L(P, f h) nd the upper Cvlieri sum U(P, f, h) is equivlent to the upper Riemnn sum U(P, f h): (.) U(P, f, h) U(P, f h). Proof. We first consider the lower sums of (.). Since the trnsformtion function h is strictl monotone, continuous nd bijective on [, b] we cn choose vlues of i to minimize the vlue of f in the intervl [ k, k ] nd so minimizing f h in the intervl [ k, k ]. The proof of (.) is similr.

12 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER Remrk.. Proposition. will be used repetedl to prove mn of the remining results for Cvlieri integrtion, since eisting results for Riemnn sums will hold trivill for the corresponding Cvlieri sums. We now give two importnt results from Riemnn integrtion theor. Proposition.5. Suppose P {,,..., n } is prtition of the closed intervl [, b], nd f bounded function defined on tht intervl. Then we hve: The lower Riemnn sum is incresing with respect to refinements of prtitions, i.e. L(P, f) L(P, f) for ever refinement P of the prtition P. The upper Riemnn sum is decresing with respect to refinements of prtitions, i.e. U(P, f) U(P, f) for ever refinement P of the prtition P. L(P, f) R(P, f) U(P, f) for ever prtition P. Proof. The proof is tken from [6]. The lst sttement is simple to prove: tke n prtition P {,,..., n }. Then inf{f(), k k } f(t k ) sup{f(), k k } where t k is n rbitrr number in [ k, k ] nd k,,..., n. Tht immeditel implies tht L(P, f) R(P, f) U(P, f). The other sttements re somewht trickier. In the cse tht one dditionl point t is dded to prticulr subintervl [ k, k ], let c k sup f() in the intervl [ k, k ], A k sup f() in the intervl [ k, t ], B k sup f() in the intervl [, k ]. Then c k A k nd c k B k so tht: c k ( k k ) c k ( k t + t k ) c k ( k t ) + c k (t k ) B k ( k t ) + A k ( t k ), which shows tht if P {,,..., k, k,..., n } nd P {,,..., k, t, k,..., n } then U(P, f) U(P, f). The proof for generl refinement P of P uses the sme ide plus n elborte indeing scheme. No more detils should be necessr. The proof for the sttement regrding the lower sum is nlogous. Proposition.6. Let f be continuous on [, b]. Then there is unique number I stisfing L(P, f) I U(P, f) for ever prtition P of [, b]. Proof. The proof is tken from []. From Proposition.5 it follows tht ever lower sum of f on [, b] is less thn or equl to ever upper sum. Thus the collection L of ll lower sums is bounded bove (b n upper sum) nd the collection U of ll upper sums is bounded below (b n lower sum). B the Lest Upper Bound Aiom, L hs lest upper bound L nd U hs gretest lower bound G. From our preceding remrks it follows tht L(P, f) L G U(P, f) for ech prtition P of [, b]. Moreover, n number I stisfing for ech prtition P of [, b] must stisf L(P, f) I U(P, f) L I G

13 CAVALIERI INTEGRATION since L is the lest upper bound of the lower sums nd G is the gretest lower bound of the upper sums. Hence to complete the proof of the theorem it is enough to prove tht L G. Let ɛ >. Since f is continuous on [, b], it follows tht f is uniforml continuous on [, b]. Thus there is δ > such tht if nd re in [, b] nd < δ, then f() f() < ɛ b. Let P {,,..., n } be prtition of [, b] such tht k < δ for k n, nd let M k nd m k be, respectivel, the lrgest nd smllest vlues of f on [ k, k ]. Then U(P, f) L(P, f) M k k m k k k k (M k m k ) k k < ɛ k b k ɛ (b ) b ɛ. Since L(P, f) L G U(P, f), it follows tht G L U(P, f) L(P, f) ɛ. Since ɛ ws rbitrr, we conclude tht L G. Definition.7 (Definite Riemnn integrl). Let f be continuous on [, b]. The definite Riemnn integrl of f from to b is the unique number I stisfing L(P, f) I U(P, f) for ever prtition P of [, b]. This integrl is denoted b f() d. We now stte (nd prove) the equivlent of Proposition.5 for lower nd upper Cvlieri sums: Proposition.8. We clerl hve: The lower Cvlieri sum is incresing with respect to refinements of prtitions, i.e. L(P, f, h) L ( P, f, h) for ever refinement P of the prtition P. The upper Cvlieri sum is decresing with respect to refinements of prtitions, i.e. U(P, f, h) U ( P, f, h) for ever refinement P of the prtition P. L(P, f, h) C(P, f, h) U(P, f, h) for ever prtition P. Proof. The proof follows trivill from Proposition. nd Proposition.5 (since ever Cvlieri sum corresponds to n equivlent Riemnn sum). Proposition.9. Let f be continuous on [, b ]. Then there is unique number I stisfing L(P, f, h) I U(P, f, h) for ever prtition P of [, b]. Proof. The proof follows trivill from Proposition. nd Proposition.6.

14 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER We cn now finll define the Cvlieri integrl: Definition. (Definite Cvlieri integrl). Let f be continuous on [, b ]. The definite Cvlieri integrl of f() from to b() is the unique number I stisfing L ( P, f, h) I U(P, f, h) for ever prtition P of [, b]. This integrl is denoted b f() d. Definition.. Let R be n Cvlieri region s given in Definition. then the re A of the region R is defined to be A f() d. Proposition.. The following Cvlieri nd Riemnn integrls re equivlent: f() d f h() d. Proof. B noting tht L(P, f h) L(P, f, h) I U(P, f, h) U(P, f h), the proof follows trivill from Proposition., Proposition.6 nd Proposition.9. Theorem.. For n ɛ > there is number δ > such tht the following sttement holds: If n subintervl of P hs length less thn δ, nd if k t k k for ech k between nd n, then the ssocited Cvlieri sum n k f(t k ) k stisfies. f() d f(t k) k < ɛ. k Proof. This proof ws dpted from []. For n ɛ > choose δ > such tht if nd re in [, b ] then < δ, then f() f() < ɛ b. If P is chosen so tht k < δ for ech k, then b Proposition.9, U(P, f, h) L(P, f, h) ɛ. Moreover, if k t k k for k n, then It follows tht Since L(P, f, h) we conclude tht m k k k L(P, f, h) m k f(t k) M k. f(t k) k k f() d M k k U(P, f, h). k f() d U(P, f, h), f(t k) k < ɛ. k

15 CAVALIERI INTEGRATION 5 B combining Proposition. nd Theorem. we finll hve f() d lim n k lim n k f(t k) k f h(t k) k f h() d, where the lst line follows from the well known fct tht the limit of Riemnn sum equls the Riemnn integrl... The Cvlieri integrl s Riemnn-Stieltjes integrl. When evluting Cvlieri integrl from to b(), it m sometimes be more convenient to consider n equivlent Riemnn-Stieltjes integrl from to b thn the ordinr Riemnn integrl from to b. To trnsform the Cvlieri integrl into n equivlent Riemnn-Stieltjes integrl, we will mke use of the inverse trnsformtion function g : h (which is gurnteed to eist, since h is bijective function). Definition. (Inverse trnsformtion function g). Let be trnsltionl function. The mpping g : [, b ] [, b], which mps i [, b ] to i [, b], is defined s g( i ) : i f( i ) +, which we will refer to s the inverse trnsformtion function. Proposition.5. The following Cvlieri nd Riemnn-Stieltjes integrls re equivlent: Proof. From Theorem. we hve (.) f() d f() dg(). n f() d lim f( i ) i. n B noting tht i i+ i g( i+ ) g( i ), nd tht g( ) nd g(b ) b, we cn re-write (.) s i n f() d lim f( i ) [ g( i+) g( i ) ], n i which we recognize s the Riemnn-Stieltjes integrl f() dg(), s required. Whenever g is differentible, we cn convenientl epress the Cvlieri integrl simpl in terms of f() nd : [ f() d f() d f() df() ] d. d d

16 6 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER. Cvlieri Integrtion: Worked Emples Severl full worked emples of Cvlieri integrtion re given below. We first present simple emple of Cvlieri integrtion from first principles (Emple.), followed b the integrtion of Cvlieri region in which f() is nonliner (Emple.). In Emple. the boundr functions re lso nonliner, followed b Emple. in which the boundr function b() is no longer required to be trnsltion of. In Emple.5 we show tht the trnsformtion function h cn be fiendishl difficult to find, but we show tht the Riemnn-Stieltjes formultion leds to much simpler solution in Emple.6. In Emple.7 we show tht the Cvlieri integrl cn be used to integrte non-cvlieri regions (with non-trnsltionl), nd in Emple.8 we show tht the trnsformtion function h cn be strictl decresing. Finll, in Emple.9 we show tht the Cvlieri integrl cn be used in some instnces where the function f() is not even defined. Emple. (Cvlieri integrtion from first principles; f(), nd b() liner). Consider the Cvlieri region bounded b the -is nd the lines f(),, nd b(). This region is shown in Figure. f() b() b Figure. Region bounded b the -is nd the lines f(),, nd b(). Also consider prtition ( i )n i on the -is such tht < < < n b, nd i i+ i. We cn form the Cvlieri integrl (using the left hnd rule) s follows: (.) n f() d lim f( i ) i. n i The prtition points i s used in the Cvlieri sum is shown in Figure. To trnsform the Cvlieri sum given in (.) into n ordinr Riemnn sum, we must find n epression for i in terms of the prtition points i, for ll i,,..., n. First consider the collection of functions { + [ i ] i : i,,..., n}. To find the prtition points i in terms of i we substitute the function

17 CAVALIERI INTEGRATION 7 + [ ] + [ ] + [ ] () n (b) n Figure. Prtition points i s used in the Cvlieri sum. f( i ) for to obtin: f( i ) + [ i ] i i + i i i i, so tht we hve the generl epression i h( i ), with h() /. Finll this llows us to rewrite the Cvlieri integrl from (.) s n equivlent Riemnn integrl: (.) n f() d lim f( i ) i n i n lim f h( i ) i n i f h() d. Evluting the Riemnn integrl of (.) with nd b we obtin f() d f h() d.75, d

18 8 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER which we cn quickl verif to be correct b evluting the re of the region shown in Figure with ordinr Riemnn integrtion: d + d d d.75 f() d. Emple. (Cvlieri integrtion; f() nonliner). Consider the Cvlieri region bounded b the -is nd the functions f(),, nd b(). This region is shown in Figure, long with the strips of integrtion. f() b() Figure. Region bounded b the -is nd the functions f(),, nd b(). The re of this region cn be clculted with the Cvlieri integrl (.) f() d f h() d. To evlute (.) we first need to find h using Definition.: f( i ) + [ i ] i ( i ) + + i i ( i ) + i i i ) ( i + h( i ).

19 CAVALIERI INTEGRATION 9 We cn now clculte (.) with h() ( + ) s follows f() d f h() d ( + ) d 96 ( + )( ) 9 + ( ).96. One cn lso compute the re under considertion (see Figure ) using ordinr Riemnn integrtion: ( 7 ) ( 5 ) d + d ( 7 ) ( 5 ) d d.96 f() d. Emple. (Cvlieri integrtion; f(), nd b() nonliner). Consider the Cvlieri region bounded b the -is nd the functions f(),, nd b(). This region is shown in Figure 5, long with the strips of integrtion. f() b() b Figure 5. Region bounded b the -is nd the functions f(),, nd b(). The re of this region cn be clculted with the Cvlieri integrl (.) f() d f h() d.

20 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER To evlute (.) we first need to find h: f( i ) + [ i ] i i + + i i i i h( i ). We cn now clculte (.) with h() / s follows f() d f h() d. d We cn once gin verif our nswer bove b computing the re of the region shown in Figure 5 with ordinr Riemnn integrtion: d + ( ) d d ( ) d f()d. Emple. (Cvlieri integrtion; b() not trnsltion of ). Consider the Cvlieri region bounded b the -is nd the functions f(),, nd b(). This region is shown in Figure 6. b() f() b Figure 6. Region bounded b the -is nd the functions f(),, nd b(). To obtin the shded re bounded in Figure 6 we will subtrct the two Cvlieri integrls shown in Figure 7 nd Figure 8. Tht is, we will compute the desired re b evluting A B.

21 CAVALIERI INTEGRATION f() A b Figure 7. Region bounded b the -is nd the functions f() nd b(). (.5) (.6) The re of A cn be clculted with the Cvlieri integrl f() d To evlute (.5) we first need to find h : f( i ) + i i i + i i i + i i i ( i + h ( i ). f h () d. ) i + We cn now clculte (.5) with h () ( +.5).5 + s follows: f() d f h () d ( +.5).5 + d ( + ) The re of B cn be clculted with the Cvlieri integrl (see Figure 8) f() d To evlute (.6) we first need to find h : (f( i )) + i i i + i i f h () d. i i

22 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER f() B b Figure 8. Region bounded b the -is nd the functions f() nd. We cn now clculte (.6) with h () / s f() d. f h () d.5 d Finll we obtin the desired re A B f() d f() d Emple.5 (Cvlieri integrtion; h difficult to find). Consider the Cvlieri region bounded b the -is nd the functions f(),, nd b(). This region is shown in Figure 9, long with the strips of integrtion. f() b() (.7) Figure 9. Region bounded b the -is nd the functions f(),, nd b(). The re of this region cn be clculted with the Cvlieri integrl f() d f h() d.

23 CAVALIERI INTEGRATION To evlute (.7) we first need to find h: f( i ) + [ i ] i ( i ) + + i i ( i ) + i i. Solving for i in terms of i produces h() which is equl to: (.8) h() with G() G() G() G() We cn now clculte (.7) with h() given b (.8) s follows: f() d f h() d (.669, ) G() G() d G() which we will once gin verif b using ordinr Riemnn integrtion: d + d d d.669 f() d. Emple.6 (Riemnn-Stieltjes formultion). Consider the Cvlieri region bounded b the -is nd the functions f(),, nd b(). This region is shown in Figure, long with the strips of integrtion. Note tht this is the sme region s studied in Emple.. We will show tht the Riemnn-Stieltjes formultion is considerbl simpler thn the direct method in which we need to find h eplicitl. The re of this region cn be clculted with the Cvlieri integrl (.9) f() d f() dg(). To evlute (.9) we first need to find g using Definition.: i i f( i ) + ( i ) + i g( i ).

24 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER f() b() Figure. Region bounded b the -is nd the functions f(),, nd b(). We cn now clculte (.9) with g() + s follows f() d f() dg() f()g () d ( 7 ) ( 5 ) +.96, which is the sme s obtined in Emple.. ( + ) d ( 7 ) ( 5 ) Emple.7 (Cvlieri integrtion; non-trnsltionl). Consider the non- Cvlieri region R shown in Figure.A: (, ) (, ) (, ) f() f() f() R A. R b() + A R + d B. R b() + A R + d C. Figure. The region bounded b f() nd.

25 CAVALIERI INTEGRATION 5 The re of this region cn be clculted with the double integrl: nd lso with the integrl: A R d d d d, A R d. We cn lso clculte the re A R with the difference between two Cvlieri integrls. The two res being subtrcted re shown in Figure.B nd Figure.C. A R A R A R +. + f h () d ( + f h () d ) d ( ) d Emple.8 (Cvlieri integrtion; h strictl decresing). Consider the Cvlieri region bounded b the -is nd the functions f(),, nd b(). This region is shown in Figure. The re of this region cn be clculted with the Cvlieri integrl (.) f() d To evlute (.) we first need to find h: so tht h(). f( i ) + [ i ] i f h() d. i i,

26 6 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER f() b() Figure. Region bounded b the -is nd the functions f(),, nd b(). We cn now clculte (.) s follows f() d f h() d d. Emple.9 (Cvlieri integrtion; f() not defined). Consider the norml Riemnn integrtion tsk given below (shown in Figure ):.5 d.5 d d.5.5 d...5 b().5.5 Figure. Region bounded b the -is, the line nd the functions.5 nd b(). Note tht the shded region in Figure is not Cvlieri region, since f() is not even defined. Nevertheless, we cn compute this re s single Cvlieri integrl s follows. We find the trnsformtion function h b equting ( i ) + [ i.5] i, i where the i on the right hnd side is set to zero since the region is bounded from the left b, nd the i on the left hnd side remins unchnged, since we re

27 CAVALIERI INTEGRATION 7 rell interested in the -intercepts of ech trnsltion of. Therefore we find i h( i ) i, so tht we cn compute the shded re s ( ) d.5.. h() d d 5. Conclusion We hve presented novel integrl f() d in which non-rectngulr integrtion strips were used. We lso presented two methods of evluting Cvlieri integrls b estblishing the following reltionships between Cvlieri, Riemnn nd Riemnn-Stieltjes integrls: f() d which is equivlent to noting tht s shown in Figure. f h() d Are A Are B Are C, f() dg(), f A f h B b b b fg re A re B re C C b Figure. Reltionships between Cvlieri (region A), Riemnn (region B) nd Riemnn-Stieltjes (region C) integrls. The reson for clling f() d the Cvlieri integrl should now become trnsprentl cler: the re of region B is equl to the re of region A b Cvlieri s principle.

28 8 T. L. GROBLER, E. R. ACKERMANN, A. J. VAN ZYL #, AND J. C. OLIVIER References [] K. Andersen, Cvlieri s Method of Indivisibles, Arch. Hist. Ect Sci., vol., no., pp. 9 67, 985. [] R. Ellis nd D. Gulick, Clculus with nltic geometr. (th ed.), Hrcourt Brce Jovnovich College Publisher, Orlndo, Florid, 99, pp. A- A-. [] H. Eves, Two Surprising Theorems on Cvlieri Congruence, College Mth. J., vol., no., pp. 8, 99. [] J. J. O Connor nd E. F. Robertson, Antiphon the Sophist, 999. [Online]. Avilble: http: //www-histor.mcs.st-ndrews.c.uk/biogrphies/antiphon.html [5] A. A. Ruff, The generlized method of ehustion, Int. J. Mth. Mth. Sci., vol., no. 6, pp. 5 5,. [6] B. G. Wchsmuth, Interctive Rel Anlsis, 7. [Online]. Avilble: org/nlsis/rels/integ/proofs/rsums.html Deprtment of Electricl, Electronic nd Computer Engineering, Universit of Pretori, nd Defence, Pece, Sfet nd Securit; Council for Scientific nd Industril Reserch, Pretori, South Afric. E-mil ddress: trienkog@gmil.com Deprtment of Electricl, Electronic nd Computer Engineering, Universit of Pretori, nd Defence, Pece, Sfet nd Securit; Council for Scientific nd Industril Reserch, Pretori, South Afric. E-mil ddress: etienne.ckermnn@ieee.org # Deprtment of Mthemtics nd Applied Mthemtics, Universit of Pretori, Pretori, South Afric. E-mil ddress: gusti.vnzl@up.c.z Deprtment of Electricl, Electronic nd Computer Engineering, Universit of Pretori, nd Defence, Pece, Sfet nd Securit; Council for Scientific nd Industril Reserch, Pretori, South Afric. E-mil ddress: jc.olivier@up.c.z