A Lost Theorem: Definite Integrals in Asymptotic Setting
aa r X i v : . [ m a t h . C A ] A p r A Lost Theorem: Definite Integrals in AnAsymptotic Setting
Ray Cavalcante and Todor D. Todorov
We present a simple yet rigorous theory of integration that is based on two axiomsrather than on a construction involving Riemann sums. With several exampleswe demonstrate how to set up integrals in applications of calculus without usingRiemann sums. In our axiomatic approach even the proof of the existence of thedefinite integral (which does use Riemann sums) becomes slightly more elegantthan the conventional one. We also discuss an interesting connection between ourapproach and the history of calculus. The article is written for readers who teachcalculus and its applications. It might be accessible to students under a teacher’ssupervision and suitable for senior projects on calculus, real analysis, or history ofmathematics.Here is a summary of our approach. Let ρ : [ a, b ] → R be a continuous functionand let I : [ a, b ] × [ a, b ] → R be the corresponding integral function, defined by I ( x, y ) = Z yx ρ ( t ) dt. Recall that I ( x, y ) has the following two properties: (A) Additivity : I ( x, y ) + I ( y, z ) = I ( x, z ) for all x, y, z ∈ [ a, b ]. (B) Asymptotic Property : I ( x, x + h ) = ρ ( x ) h + o ( h ) as h → x ∈ [ a, b ],in the sense that lim h → I ( x, x + h ) − ρ ( x ) hh = 0 . In this article we show that properties (A) and (B) are characteristic of thedefinite integral. More precisely, we show that for a given continuous ρ : [ a, b ] → R ,there is no more than one function I : [ a, b ] × [ a, b ] → R with properties (A) and (B) .This will justify the simple definition R ba ρ ( x ) dx def = I ( a, b ), where I ( x, y ) is a functionsatisfying (A) and (B) . In this manner, we are able to rigorously develop the theoryof definite integrals and their applications without partitioning the interval [ a, b ] andwithout using Riemann sums. Notice that, at this stage, the existence of the integralis still unsettled. Next, we prove that if R ( x ) is an antiderivative of ρ ( x ), then I ( a, b ) = R ( b ) − R ( a ). Conventionally, this formula is used for explicit evaluation, ut in our approach it also guarantees the existence of the integral for all ρ ( x )with an explicit antiderivative R ( x ). Our approach utilizes the Riemann partitionprocedure for only one purpose : to prove the existence of the definite integral forfunctions without explicit anti-derivatives such as f ( x ) = e − x . Also, our proof(Theorem 3.2) seems to be slightly more elegant than the conventional one. Nextwe use axioms similar to (A) and (B) to define the concepts of area under thecurve, arclength, volumes of revolution, etc. More precisely, we define all of thesegeometrical quantities as being additive and asymptotically equal to their Euclideancounterparts (such as the area of a rectangle, the Euclidean distance between twopoints, the volume of a cylindrical shell, etc.). Our definitions are mathematicallycorrect and well motivated. Derivations of the corresponding integral formulas (forthe area under a curve, arclength, volume, etc.) appear in our approach as simplerigorous theorems with proofs done in the spirit of asymptotic analysis; we involveneither partitions of the interval [ a, b ] nor Riemann sums.The elementary theory of integration presented in this article (and summarizedabove) opens the door for a simple yet completely rigorous method of teachingintegration and its applications in a calculus course or a beginning real analysiscourse. Also, we strongly recommend our method of setting up integrals (withoutRiemann sums) for teaching physics and engineering courses based on calculus.The method presented in this article has a long and interesting history. Thereader might be surprised to learn that practically all textbooks on calculus and itsapplications that were written in the period between Leibniz and Riemann motivate,define, and set up integrals by methods very similar to the method presented in thisarticle, although disguised in the language of infinitesimals. The method (amongother treasures) was lost in the history of calculus after infinitesimals were abolishedat the end of 19th century. This explains the choice of the title “A Lost Theorem....”We shall briefly discuss the connection of our approach with infinitesimal calculusin Section 5. In the modern literature we identify three sources using methods forintegration similar to our approach. In H. J. Keisler’s calculus textbook [4] (lookfor Infinite Sum Theorem on p. 303), the reader will find a method similar to oursin the framework of nonstandard analysis (in a very accessible form). We also referto S. Lang ([5], pp. 292-296), L. Gillman and R. McDowell ([2], pp. 176-179) and L.Gillman [3], where a property similar to (B) is used for a definition of the definiteintegral.
For our axiomatic approach to integration we assume knowledge of limits, continu-ity, and derivatives at the level of a typical beginning calculus course. In contrast,we do not assume any knowledge of integration. Finally, we need several elementaryconcepts and notation borrowed from asymptotic analysis, which we present in thissection. Most of the results are elementary, and we leave the proofs to the reader.
Definition 2.1
We denote by o ( x n ) the set of all functions f : D f → R such that D f ⊆ R , ∈ D f , and lim x → f ( x ) /x n = 0, where D f stands for the closure of f in R . We summarize this as o ( x n ) = { f ( x ) : f ( x ) /x n → → } . It iscustomary to write f ( x ) = o ( x n ) instead of the more precise f ∈ o ( x n ) in the casewhen f is an unspecified function in o ( x n ) . If n = 0, the above definition reducesto f ( x ) = o (1) if f ( x ) → x → Example 2.1 x = o ( x ) since x /x → x →
0. In contrast, sin x = o ( x ) sincesin x/x → x →
0. Also, we have sin x = o (1) since sin x → x → Theorem 2.1 (Increment Theorem)
Let f : [ a, b ] → R be a function and x ∈ ( a, b ) . Then f is differentiable at x if and only if f ( x + h ) − f ( x ) = f ′ ( x ) h + o ( h ) . Lemma 2.1
Let f ( x ) = o ( x m ) and g ( x ) = o ( x n ) . Then (a) f ( x ) ± g ( x ) = o ( x k ) where k = min( m, n ) , and (b) f ( x ) g ( x ) = o ( x m + n ) . Also, (c) if f ( x ) = o (1) and g ( x ) = o (1) and f is continuous at , then f ( g ( x )) = o (1) . Remark 2.1 (Asymptotic Algebra)
It is customary in asymptotic analysis towrite simply o ( x m ) ± o ( x n ) = o ( x k ), o ( x m ) ± o ( x n ) = o ( x m + n ) and o ( o (1)) = o (1)instead of (a), (b), and (c) in the above lemma, respectively, when no confusioncould arise. We shall use this notation. Lemma 2.2 (Absolute Value)
Let A ∈ R . Then | A + o ( x n ) | = | A | + o ( x n ) . Lemma 3.1 (Uniqueness)
Let a, b ∈ R , a < b , and let ρ : [ a, b ] → R be acontinuous function. Let I : [ a, b ] × [ a, b ] → R be a function in two variablessatisfying the axioms (A) and (B) at the beginning of the introduction. Then I ( a, x ) satisfies the initial value problem ddx I ( a, x ) = ρ ( x ) , I ( a, a ) = 0 on the interval ( a, b ) . Consequently, there is no more than one function I : [ a, b ] × [ a, b ] → R satisfying the properties (A) and (B) . Proof:
Suppose that I ( x, y ) satisfies (A) and (B) . We have ddx I ( a, x ) = lim h → I ( a, x + h ) − I ( a, x ) h (A) = lim h → I ( x, x + h ) h =lim h → I ( x, x + h ) − ρ ( x ) h + ρ ( x ) hh = lim h → I ( x, x + h ) − ρ ( x ) hh + ρ ( x ) (B) = 0 + ρ ( x ) = ρ ( x ) . (1) Also, (A) implies I ( a, a ) + I ( a, b ) = I ( a, b ), so I ( a, a ) = 0. Suppose that J ( x, y ) is another function satisfying the axioms (A) and (B) , and let ∆( x, y ) = I ( x, y ) − J ( x, y ). We have ddx ∆( a, x ) = 0 and ∆( a, a ) = 0, and hence ∆( a, x ) = 0for all x in ( a, b ). Consequently, for a given ρ ( x ) there is no more than one I ( a, x ). ext, (A) implies I ( x, y ) = I ( a, y ) − I ( a, x ). Thus I ( a, x ) uniquely determines I : [ a, b ] × [ a, b ] → R . N Notice that the uniqueness result presented above does not involve partitioningof the interval [ a, b ] and Riemann sums. Rather, it is based on the more elementaryresult from calculus that “every function with zero derivative on an interval is aconstant.”The above lemma justifies the following definition:
Definition 3.1 (Axiomatic Definition)
Let ρ : [ a, b ] → R be a continuous func-tion and let I : [ a, b ] × [ a, b ] → R be a function in two variables satisfying the axioms (A) and (B) at the beginning of the introduction. Then the value I ( a, b ) is calledthe integral of ρ ( x ) over [ a, b ]. We shall use the usual notation R ba ρ ( x ) dx def = I ( a, b ).Note that axioms (A) and (B) in the beginning of the introduction are easilymotivated and visualized (see Figure 1). We observe as well that if a ≤ α ≤ β ≤ a ,then the restriction I ↾ [ α, β ] × [ α, β ] also satisfies the axioms (A) and (B) , andthus R βα ρ ( x ) dx = I ( α, β ). Theorem 3.1 (Fundamental Theorem)
As before, let ρ ( x ) be a continuous func-tion on [ a, b ] . (i) If the integral I ( a, x ) of ρ ( t ) over [ a, x ] exists for every x in [ a, b ] , then ddx I ( a, x ) = ρ ( x ) on ( a, b ) . (ii) If R ( x ) is an anti-derivative of ρ on [ a, b ] , then I ( a, b ) = R ( b ) − R ( a ) . Proof:
Part (i) follows directly from Lemma 3.1. For (ii), we have ddx I ( a, x ) = ρ ( x )by (i) and ddx R ( x ) = ρ ( x ) by assumption. It follows that ddx [ I ( a, x ) − R ( x )] = 0 on( a, b ), implying R ( x ) = I ( a, x ) + C for some constant C . Hence, R ( b ) − R ( a ) =[ I ( a, b ) + C − ( I ( a, a ) + C )] = I ( a, b ), as required, since I ( a, a ) = 0 by Lemma 3.1. N Notice that our simple theory assumes that the function I ( x, y ), and therebythe integral I ( a, b ), exists. The following is our first existence result (for the generalexistence result see Theorem 3.2). Corollary 3.1 (Weak Existence) If ρ ( x ) has an antiderivative on [ a, b ] , thenthe integral of ρ ( x ) over [ a, b ] exists. Proof:
Let R ( x ) be an antiderivative of ρ ( x ). Then the function I : [ a, b ] × [ a, b ] → R defined by I ( x, y ) = R ( y ) − R ( x ) satisfies (A) and (B) (at the beginning of theintroduction) and it is the only function which satisfies (A) and (B) by Lemma 3.1.The number I ( a, b ) is the integral we are looking for. N Remark 3.1 (Basic Properties of the Integral)
The basic properties of theintegral follow immediately from part (ii) of Theorem 3.1 under the assumptionthat the integrals exist. For example, Theorem 3.1 implies the linear property R [ c f ( x ) + c g ( x )] dx = c R f ( x ) dx + c R g ( x ) dx provided that at least two of thethree integrals exist. ur simple yet rigorous theory presented so far is powerful enough to supportmost of the topics related to the integral and its applications in a typical beginningcalculus course. To deal with integrals such as R ba e − x dx and R ba sin ( x ) dx , we needa general existence result. This (and only this) is the place in our approach wherewe use partitions of the interval [ a, b ] and Riemann sums.Let [ a, b ] be, as before, a closed interval in R and x, y ∈ R , a ≤ x < y ≤ b . Recallthat a partition of [ x, y ] is a finite ordered set of the form P = { x , x , . . . , x n } ,where n ∈ N and x = x < x < · · · < x n = y . We denote by P [ x, y ] theset of all partitions of [ x, y ]. Let ρ : [ a, b ] → R be a continuous function and P = { x , x , . . . , x n } be a partition of [ x, y ]. Recall that L ( P ) = n X k =1 (cid:18) min x k − ≤ t ≤ x k ρ ( t ) (cid:19) ( x k − x k − ) ,U ( P ) = n X k =1 (cid:18) max x k − ≤ t ≤ x k ρ ( t ) (cid:19) ( x k − x k − ) , are called the lower and upper Darboux sums of ρ ( t ) determined by P , respectively.Let x, y, z ∈ R , a ≤ x < y < z ≤ b . Notice that if P ∈ P [ x, y ] and Q ∈ P [ y, z ],then P ∪ Q ∈ P [ x, z ] and we have L ( P ) + L ( Q ) = L ( P ∪ Q ) and U ( P ) + U ( Q ) = U ( P ∪ Q ). The next result can be found in any contemporary textbook on Riemannintegration. Lemma 3.2
Let ρ : [ a, b ] → R be a continuous function. Then: (i) For every two partitions P and Q of [ a, b ] we have (min [ a,b ] ρ )( b − a ) ≤ L ( P ) ≤ U ( Q ) ≤ (max [ a,b ] ρ )( b − a ) . (ii) For every ǫ ∈ R + there exists a partition P of [ a, b ] such that U ( P ) − L ( P ) <ǫ . Theorem 3.2 (General Existence Result)
Let ρ : [ a, b ] → R be a continuousfunction. Then ρ has both an integral and an antiderivative. Proof:
Suppose, first, that x, y ∈ R , a ≤ x < y ≤ b . We observe that the set { L ( P ) | P ∈ P [ x, y ] } is bounded from above by the number (max x ≤ t ≤ y ρ ( t ))( y − x ).Thus I ( x, y ) = sup { L ( P ) | P ∈ P [ x, y ] } exists in R by the completeness of R . Weintend to show that I ( x, y ) satisfies the axioms (A) and (B) at the beginning ofthe introduction. We start with ( B ): we have (min x ≤ t ≤ y ρ ( t ))( y − x ) ≤ I ( x, y ), bythe definition of I ( x, y ), since P = { x, y } is a partition of the interval [ x, y ]. We let y − x = h and the result is (min x ≤ t ≤ x + h ρ ( t )) h ≤ I ( x, x + h ) ≤ (max x ≤ t ≤ x + h ρ ( t )) h .The latter implies axiom (B) since ρ is continuous at x . To prove ( A ), we observethat L ( P ) ≤ I ( x, y ) ≤ U ( P ) for every partition P ∈ P [ x, y ]. Indeed, the firstinequality follows directly from the definition of I ( x, y ) and the second inequalityfollows from the definition of I ( x, y ) and part (i) of Lemma 3.2. Next, suppose hat x, y, z ∈ R , a ≤ x < y < z ≤ b , and let Q ∈ P [ y, z ]. As before we have L ( Q ) ≤ I ( y, z ) ≤ U ( Q ), and also L ( P ∪ Q ) ≤ I ( x, z ) ≤ U ( P ∪ Q ) since P ∪ Q is apartition of [ x, z ]. After summing up we obtain: L ( P ) + L ( Q ) − U ( P ∪ Q ) ≤ I ( x, y ) + I ( y, z ) − I ( x, z ) ≤ U ( P ) + U ( Q ) − L ( P ∪ Q ) . Now we can choose partitions P ∈ P [ x, y ] and Q ∈ P [ y, z ] such that U ( P ) − L ( P ) <ǫ/ U ( Q ) − L ( Q ) < ǫ/ U ( P ∪ Q ) − L ( P ∪ Q ) < ǫ since L ( P )+ L ( Q ) = L ( P ∪ Q ) and U ( P )+ U ( Q ) = U ( P ∪ Q ). Thus the above chainof inequalities reduces to − ǫ < I ( x, y ) + I ( y, z ) − I ( x, z ) < ǫ , implying I ( x, y ) + I ( y, z ) = I ( x, z ), as required. Finally we can eliminate the restriction on x, y and z by extending the function I ( x, y ) to a function in the form I : [ a, b ] × [ a, b ] → R by letting I ( x, y ) = − I ( y, x ) and I ( x, x ) = 0 for all x and y in [ a, b ]. Notice thatthe function I ( x, y ) just defined also satisfies (A) and (B) ; thus it is uniquelydetermined, by Lemma 3.1. The number I ( a, b ) is the integral we are looking for,by Definition 3.1, and R ( x ) = I ( a, x ) is an antiderivative of ρ ( x ), by part (i) ofTheorem 3.1. N Imagine that you are a young instructor preparing to cover arclength in a typicalcalculus course. We can safely assume that before going in front of your studentsyou would like to clarify the structure of the topic for yourself: what is the definitionof “arclength,” how do I motivate it, is there a theorem to present (with or withoutformal proof), and finally which examples should I use? One option is to define theconcept of arclength directly by the integral formula L ( a, b ) = R ba p f ′ ( x ) dx . Ifthis is your choice, your next task will be to motivate this definition. You might usethe integral formula to calculate some familiar results from high school mathemat-ics: the distance formula for a line segment or the formula for the circumference ofa circle. This approach, although completely legitimate, is rarely used by calculustextbooks. The integral formula still looks terribly unmotivated even after derivingthe distance formula for a line segment. Also, it is far from clear that this is theonly formula producing the distance formula. For that reason most calculus booksuse Riemann sums to convince students that the integral formula is “reasonable.”What follows is messy mathematics: the partition of the interval [ a, b ] gives theimpression that the step of the partition ∆ x is “very small” which leads to theconclusion that ∆ L is approximately equal to p ∆ x + ∆ y . The latter producesour integral formula after factoring out ∆ x and taking the limit as ∆ x →
0. Youmight ponder for hours questions such as: a) What, after all, is the definition of“arclength”? Is the integral formula exact or approximate ? After all, its derivationinvolves the approximate formula ∆ L ≈ p ∆ x + ∆ y . b) What is meant by “∆ x is very small”? If ∆ x = 0, then ∆ y = 0 and the root p ∆ x + ∆ y is also equal tozero. If ∆ x = 0, then ∆ L = p ∆ x + ∆ y (unless the curve is a straight line). c) Is he “derivation” of the formula for arclength a sort of casually presented proof of acasually stated theorem? And if yes, what is the rigorous version of this theorem?Worst of all is that the students are usually preoccupied with the technicalities ofthe partition procedure and the sigma notation in the Riemann sums and hardlypay attention to the fact that a new important concept has just been introduced.In this section we take another approach. The concept of arclength is definedas an additive quantity which is asymptotically equal to the Euclidean distancebetween two points (Definition 4.2). The definition is mathematically correct andwell motivated. It is based on the concept of limit, not integral. Next we derive thearclength integral formula as a simple rigorous theorem in the spirit of asymptoticanalysis (Theorem 4.2). Similarly we define the rest of the additive quantities fromgeometry and physics such as area under a curve, volume of a solid of revolution ,etc. We involve neither a partition of the interval [ a, b ] nor Riemann sums.We start with area under a curve. Our assumption is that the reader knowswhat the area of a rectangle is but not necessarily what the area under a curve is. In particular, we do not assume that the reader necessarily knows the integralformula A ( a, b ) = R ba f ( x ) d x for the area under a curve; rather our goal is to derivethis formula starting from the more elementary concept of the area of a rectangle. Definition 4.1 (Area Under a Curve)
Let y = f ( x ) be a continuous functionon [ a, b ] such that f ( x ) ≥ x ∈ [ a, b ]. Let A : [ a, b ] × [ a, b ] → R be a functionin two variables satisfying the following two properties: (a) A ( x, y ) + A ( y, z ) = A ( x, z ) for all x, y, z ∈ [ a, b ]. (b) A ( x, x + h ) = ± R ( x, x + h )+ o ( h ) as h → ± for all x ∈ [ a, b ], where R ( x, x + h )denotes the area of the rectangle with vertices ( x, , ( x + h, , ( x + h, f ( x )),and ( x, f ( x )).The number A ( a, b ) is called (by definition) the area under the curve y = f ( x )and above the interval [ a, b ].The above definition can be easily motivated (see Figure 2).In the next theorem we derive the familiar integral formula for A ( a, b ) withoutpartitions or Riemann sums. While deriving this formula, we shall simultaneouslyprove two things: (a) the correctness of the above definition, and (b) the existenceof the area under the curve. As in the conventional approach, the integral formulaoffers a practical method for explicit evaluation. Theorem 4.1 A ( a, b ) = R ba f ( x ) dx . Consequently, the area A ( a, b ) is uniquelydetermined by the properties (a) and (b) in Definition 4.1. Proof:
We have to find the asymptotic expansion of A ( x, x + h ) in powers of h as h → h . Since R ( x, x + h ) = f ( x ) | h | , wehave A ( x, x + h ) = ± R ( x, x + h ) + o ( h ) = ± f ( x ) | h | + o ( h ) = f ( x ) h + o ( h ), and theabove formula follows directly from Definition 3.1 for ρ ( x ) = f ( x ). Conversely, it iseasy to verify that the function A : [ a, b ] × [ a, b ] → R defined by A ( x, y ) = R yx f ( t ) d t atisfies the properties (a) and (b) in the above definition, thus the number A ( a, b )is the area under the curve. N Next, we define the concept of arclength without partitions or Riemann sums.Our assumption is that the reader knows what the
Euclidean distance between twopoints is but not necessarily what the arclength along a curve is. In particular, we donot assume any knowledge about the integral formula L ( a, b ) = R ba p f ′ ( x ) dx ;our goal is to derive this formula starting from the more elementary concept ofEuclidean distance between two points. Definition 4.2 (Arclength)
Let f ∈ C [ a, b ] and let L : [ a, b ] × [ a, b ] → R be afunction in two variables satisfying the following two properties: (a) L ( x, y ) + L ( y, z ) = L ( x, z ) for all x, y, z ∈ [ a, b ]. (b) L ( x, x + h ) = ± D ( x, x + h )+ o ( h ) as h → ± for all x ∈ [ a, b ], where D ( x, x + h )is the Euclidean distance between the points ( x, f ( x )) and ( x + h, f ( x + h )).The number L ( a, b ) is called (by definition) the arclength of the curve y = f ( x )between the points ( a, f ( a )) and ( b, f ( b )).The above definition can be easily motivated (see Figure 3).In the next theorem we rigorously derive the formula for arclength withoutinvolving partitions of the interval or Riemann sums. Among other things we provecorrectness of the above definition and the existence of the arclength L ( a, b ). Theorem 4.2 L ( a, b ) = R ba p f ′ ( x ) dx . Consequently, the arc length L ( a, b ) isuniquely determined by the properties (a) and (b) in Definition 4.2. Proof:
We have to find the asymptotic expansion of L ( x, x + h ) in powers of h as h → ρ ( x ) in front of h (see Definition 3.1). Let∆ y = f ( x + h ) − f ( x ) and recall that ∆ y = f ′ ( x ) h + o ( h ) (Increment Theorem 2.1).We have: L ( x, x + h ) = ± D ( x, x + h ) + o ( h ) = ± q h + ∆ y + o ( h ) == ±| h | s (cid:18) f ′ ( x ) h + o ( h ) h (cid:19) + o ( h ) == h s f ′ ( x ) + 2 f ′ ( x ) o ( h ) hh + (cid:18) o ( h ) h (cid:19) + o ( h ) == h p f ′ ( x ) + o (1) + o ( h ) = h hp f ′ ( x ) + p f ′ ( x ) + o (1) − p f ′ ( x ) i ++ o ( h ) = h "p f ′ ( x ) + o (1) p f ′ ( x ) + o (1) + p f ′ ( x ) + o ( h ) == h hp f ′ ( x ) + o (1) i + o ( h ) = h p f ′ ( x ) + o ( h ) + o ( h ) == h p f ′ ( x ) + o ( h ) . or the coefficient in front of h we have ρ ( x ) = p f ′ ( x ), which implies ourintegral formula by Definition 3.1. Conversely, it is easy to verify that the function L : [ a, b ] × [ a, b ] → R defined by L ( x, y ) = R yx p f ′ ( t ) d t satisfies (a) and (b) inDefinition 4.2 . Thus the number L ( a, b ) is the arc length of the curve. N Next, we set up the integral for a volume of revolution about the y -axis withoutpartitions of the interval and Riemann sums. Our assumption is that the readerknows what a volume of a cylindrical shell is but not necessarily what a volume ofrevolution is. Definition 4.3 (Volume of Revolution)
Let y = f ( x ) be continuous on [ a, b ], f ( x ) ≥
0, and 0 ≤ a < b . Let V : [ a, b ] × [ a, b ] → R be a function in two variablessatisfying the following properties: (a) V ( x, y ) + V ( y, z ) = V ( x, z ) for all x, y, z ∈ [ a, b ]. (b) V ( x, x + h ) = ± U ( x, x + h ) + o ( h ) as h → ± for all x ∈ [ a, b ], where U ( x, x + h ) is the volume of the cylindrical shell obtained by revolving the rectanglewith vertices ( x, , ( x + h, , ( x + h, f ( x )) and ( x, f ( x )) about the y -axis (see Figure4). The number V ( a, b ) is called (by definition) the volume of revolution aboutthe y -axis of the curve y = f ( x ).In the next theorem we rigorously derive the familiar formula for the volume V ( a, b ), and we do so without partitions or Riemann sums. Among other things weprove the uniqueness and existence of the volume V ( a, b ). Theorem 4.3 V ( a, b ) = R ba πxf ( x ) dx . Consequently, the volume V ( a, b ) is uniquelydetermined by the properties (a) and (b) from Definition 4.3. Proof:
The volume of the cylindrical shell is U ( x, x + h ) = | π ( x + h ) − πx | f ( x ).Hence, with the help of Lemma 2.2, we have V ( x, x + h ) = ± U ( x, x + h ) + o ( h ) = ±| π ( x + h ) − πx | f ( x ) + o ( h ) = ±| π (2 xh + h ) | f ( x ) + o ( h ) = ± πxf ( x ) | h | + πf ( x ) h + o ( h ) = ± πxf ( x ) | h | + o ( h ) + o ( h ) = 2 πxf ( x ) h + o ( h ), and from Defini-tion 3.1 we have ρ ( x ) = 2 πxf ( x ). Conversely, it is easy to verify that the function V : [ a, b ] × [ a, b ] → R defined by V ( x, y ) = R yx πtf ( t ) dt satisfies the properties (a)and (b) in Definition 4.3. Thus the number V ( a, b ) is the volume of revolution. N As we explained at the end of the introduction, the method presented in this articlehas a long and interesting history. The purpose of this section is to establish a rela-tion between our method of integration and infinitesimal calculus. This section maybe helpful to those readers who are interested in reading original texts on infinitesi-mal calculus but who might not have background in modern nonstandard analysis.We should mention that the tradition of using infinitesimal arguments is still very uch alive and can be found in many contemporary texts on applied mathematics,physics, and engineering. For example, in the famous The Feynman Lectures onPhysics [1] we located about ten cases of integral formulas for additive physicalquantities derived in the spirit of infinitesimal calculus and without Riemann sums(see Volume I: pp. 13-3, 14-8, 43-2, 44-10 and 44-11, 46-6, 47-5; Volume II: pp. 3-2,38-6). For that reason we believe that our article and the discussion in this sectionin particular might also be helpful to pure mathematicians who are interested inreading texts on applied mathematics, physics, or engineering but who might feeluneasy with infinitesimal reasoning.Recall that in the period from Leibniz to Weierstrass, calculus was commonlyknown as infinitesimal calculus and was based on the hypothesis that there existnonzero infinitesimals, i.e., mysterious numbers dx with the property 0 < | dx | < /n for all n ∈ N . We should keep in mind that at that period not only the theory ofinfinitesimals but also the theory of real numbers was without rigorous foundation.So, the existence of nonzero infinitesimals should not be dismissed as nonsense. Yes,the field of the real numbers R does not have nonzero infinitesimals, but there wereno real numbers in the era of Leibniz and Euler; the real numbers were an inventionof the late 19th century and were systematically implemented in mathematics atthe beginning of 20th century.We shall discuss Leibniz-Euler infinitesimal calculus using the common “differ-ential notation”: Let y = f ( x ) , a ≤ x ≤ b be a function. In what follows, dx standsfor a new independent variable (real or infinitesimal depending on the context) and∆ y = f ( x + dx ) − f ( x ) stands for the increment of y . If f is differentiable at x ,then dy = f ′ ( x ) dx stands for the differential of y . It is clear that dx = ∆ x , but weprefer to use dx over ∆ x , thus keeping track of the fact that x is an independentand y is a dependent variable. Before proceeding further we should notice that inthe old infinitesimal calculus, and in the modern nonstandard analysis as well, thenotation dx rarely stands for a fixed infinitesimal number (as π, e, √
2, etc. stand forspecific irrational numbers). Rather, dx is usually used for an independent variableranging over a set consisting of both infinitesimal and real (standard) numbers. Todemonstrate this point we shall present the characterization of continuity used byEuler but rigorously justified in the modern nonstandard analysis: Let x be a real(standard) number in the domain of f . Then f is continuous at x if and only if f ( x + dx ) − f ( x ) is infinitesimal for every infinitesimal dx . In the manuscripts ofEuler this statement appears in a slightly more casual form: f is continuous at x ifand only if ∆ y is infinitesimal whenever dx is infinitesimal. It is clear that if both x and dx are real (standard) numbers, then ∆ y = f ( x + dx ) − f ( x ) is also a real(standard) number. Also, “real (standard) number” is a modern term. Leibniz andEuler would rather use “usual quantity” (as opposed to “infinitesimal quantity”)instead.Let us try to mimic, for example, the arguments used by Euler for settingup the integral for the arclength L ( a, b ) along the curve y = f ( x ) , a ≤ x ≤ b in a typical calculus course in the middle of the 18th century (compare with ourDefinition 4.2 and Theorem 4.2). If dx is infinitesimal, the arclength L ( x, x + dx )between the points ( x, f ( x )) and ( x + dx, f ( x + dx )) should be equal to the Euclidean istance D ( x, x + dx ) = p dx + ∆ y between the same points, up to infinitesimalsof second order relative to dx ; in symbols, L ( x, x + dx ) ≈ p dx + ∆ y (see Figure3, where h should be replaced by dx ). On the other hand, since L is an additivequantity, we have L ( x, x + dx ) = L ( a, x + dx ) − L ( a, x ). Thus L ( x, x + dx ) ≈ dL ,by the increment theorem (in its original infinitesimal form). Also, ∆ y ≈ dy , bythe increment theorem. The result is dL ≈ p dy/dx ) dx which implies thefamiliar integral formula L ( a, b ) = R ba p dy/dx ) dx . We should note that theimplication in the previous sentence has never been rigorously justified in the oldinfinitesimal calculus; among other things the goal of our article is to fill this gap.A contemporary mathematician, unless familiar with nonstandard analysis, willhave difficulty recognizing our asymptotic property (B) (at the beginning of theintroduction) using Euler’s language presented above. For those who are interestedin using nonstandard analysis we recommend H. J. Keisler [4] or T. Todorov [6],where the reader will find more literature on the subject. In what follows, however,we shall choose another path: we shall use the language of asymptotic analysis(Section 2) to relate the method of integration presented here with the method ofinfinitesimal calculus. For that purpose we suggest the following modification ofSection 3 and Section 4 of our article:1) First, instead of the letter h (used in our article so far) we shall use theoriginal Leibniz notation dx . In other words we let dx = h and we treat dx as anew real (standard) independent variable. In this notation our Asymptotic Property (B) (see the beginning of the introduction) becomes( B ′ ) I ( x, x + dx ) = f ( x ) dx + o ( dx ) a s dx → x ∈ [ a, b ] . As before, f ( dx ) = o ( dx n ) means f ( dx ) /dx n → dx → ≈ used by Euler inour previous example. Let F ( dx ) and G ( dx ) be two real functions. We say that F ( dx ) and G ( dx ) are equal up to infinitesimals of second order relative to dx , insymbols, F ≈ G , if F ( dx ) − G ( dx ) = o ( dx ) (i.e., if ( F ( dx ) − G ( dx )) /dx → dx → (B ′ ) becomes( B ′′ ) I ( x, x + dx ) ≈ f ( x ) dx for all x ∈ [ a, b ] .
3) We might stop writing dx →
0, since the symbol dx is more than suggestive.4) Now we can rewrite Section 3, replacing h by dx and the axiom (B) by ( B ′′ ).The additive property (A) (at the beginning of the introduction) does not needmodification.5) Finally, we have to replace all parts (b) in the definitions in Section 4 bytheir ( b ′′ )-counterparts in the spirit of ( B ′′ ). For example, part (b) of Definition 4.2should be replaced by:( b ′′ ) L ( x, x + dx ) ≈ ± D ( x, x + dx ) for all x ∈ [ a, b ] , and similarly for the rest of the ( b)s in Section 4.While preserving the content of the article, the modification suggested abovemakes the arguments of the old infinitesimal calculus more transparent. ) Readers who feel uncomfortable with the notation dx (and especially with dx →
0) should restore the notation h used in Section 3 and Section 4 of thisarticle. However we recommend our trick “replace h by dx ” to those readers whoare interested in reading original texts on infinitesimal calculus, but do not havebackground in the modern nonstandard analysis.Among other things our article suggests that the reasoning of mathematiciansin the era of Leibniz and Euler, as well as the reasoning of contemporary appliedmathematicians and physicists, is often more reliable and rigorous than is usuallyacknowledged by pure mathematicians. Acknowledgement:
We are thankful to our colleague Donald Hartig whomade several useful remarks on the manuscript.
References [1] R. P. Feynman, R. B. Leighton and M. Sands,
The Feynman Lectureson Physics , Addison-Wesley, Reading, MA, 1964.[2] L. Gillman and R. H. McDowell,
Calculus , W.W. Norton, New York,1973.[3] L. Gillman, An Axiomatic Approach to the Integral, this MONTHLY, (1993) 16-25.[4] H. J. Keisler,
Elementary Calculus: An Infinitesimal Approach , 2nded., Prindle, Weber and Schmidt, Boston, 1976.[5] S. Lang,
A First Course in Calculus , 5th ed., Addison-Wesley, Reading,MA, 1969.[6] Todor D. Todorov, Back to Classics: Teaching Limits through Infinites-imals,
Internat. J. Math. Ed. in Sci. Tech. (2001) 1-20. Ray Cavalcante was an undergraduate student in mathematics at thetime this article was written, and he is currently a graduate studentat Cal Poly, San Luis Obispo.Mathematics Department, California Polytechnic State University, SanLuis Obispo, CA 93407, USA ([email protected]).