The definite integral J: f(x)dx represents the area under the graph of the function y = f(x) between x =a and x = b. The standard way to define this is to partition the interval [a, b] into a finite number of subintervals, approximate the desired area by sums of areas of rectangles based on these subintervals, and then take the limit as the number of subintervals is increased. The hyperreal perspective suggests the alternative procedure of parti ·:ioning [a, b] into subintervals of infinitesimal width, in line with Leib-::liz's conception of the expression J y dx-with J as an elongated "S" for -sum" -as meaning the sum of all the infinitely thin rectangles of size 'J x dx. In order to develop this approach we will first review the standard definition of the integral that is associated with Riemann. 9.1 Riemann Sums Let f be a function that is bounded on [a,b] in R A partition of [a, b] is a finite set P = {xo, ... ,xn} with a= xo < · · · < Xn = b. Let Mi and mi be the least upper bound and greatest lower bound ofnf on [xi_1, xi], respectively, and L!xi = Xi -Xi_1. Define the • upper Riemann sum: U(f, P) = 2::::=1 Midxi; • lower Riemann sum: L(f, P) = 2::::=1 midxi; • ordinary Riemann sum: S(f, P) = 2::::=1 f(xi-t)dxi. 106 9. The Riemann Integral If M and m are the least upper bound and greatest lower bound of f on [a, b], then m(b(-a)L(f, P) S(f, P) Ui(f, P) M(b(-a). Also, by using refinements of partitions it is shown that any lower sum is less than or equal to any upper sum: L(f, P1) U!(f, P2). We say that f is Riemann integrable on [a, b] with integral I:f(x)dx if the latter is a real number equal to the least upper bound of the lower sums L(f, P) and also to the greatest lower bound of the upper sums U(f, P) taken over all partitions P of [a, b]. This holds iff (1) L(f, P) I:f(x)dx U(f, P) for all partitions P; and (2) for any real c > 0 there is a partition P with U(f, P)(-L(!, P) < c. For a given positive real .dx, let P.tlx = {xo, ... ,xn} be the partition [a, Xt], ... , [xn-2, Xn-1] of [a, b] into subintervals of equal width .dx, together with a (possibly smaller) last subinterval [xn-l, b]. This is given by taking n to be the least integer such that a+ n.dx 2: b, and Xk =a+ k.dx for(k< n. The partition P .tlx is uniquely determined by the number .dx (observe that if .dx 2:: b-a, then n = 1, and we just get P.tlx =x{a, b}). Now let U(f, .dx), L(f, .dx), S(f, .dx) be the upper, lower, and ordinary Riemann sums for this partition. These quantities can be regarded as functions of the real variable .dx, defined on JR+. Hence these functions extend automatically to *JR+. In particular, they are defined for all positive infinitesimals, giving a hyperreal meaning to the notion of Riemann sums for infinitesimal width partitions. For instance, we may informally think of S(f, .dx) as being the "sum" f(xo)Llx + f(xt)Llx + · · · + f(xn-2).dx + /(Xn-t)(b(-Xn-d, where n is the least hyperinteger such that a+ n.dx 2: b. When .dx is infinitesimal, this n will be unlimited. (In Section 12.7 we will analyse this informal view further, and express S(f, .dx) as a "hyperfinite sum" over a "hyperfinite partition"n.) The relationships m(b(-a) L(f, .dx) S(f, .dx) U!(f, Llx) M(b(-a) hold for all .dx E JR+, and hence by universal transfer hold for all positive hyperreal .dx, including positive infinitesimals. Similarly we have L(f, .dx) U(f, .dy) for all .dx, .dy E *JR+. 9.1 Riemann Sums 107 Theorem 9.1.1 Iff is continuous on the real interval [a, b], then for any ix positive infinitesimal L1x, Lx(f, L1x) ::::: U(f, L1x). Proof The key to this is the fact that We will find an upper bound of the right side of this equation that has the form [/(c)x-f(d)](bx-a), where c, dnare two numbers that are smaller than L1x. When L1x is infinitesimal, the continuity of f then ensures that f (c) ::::: f (d), making our upper bound infinitesimal. By a method that is now becoming familiar, we first formalise the real version of this construction, and then apply transfer. For positive real L1x, let J-L( L1x) be the maximum of the numbers Mi -mi for 1 :::; i :::; n in the partition determined by L1x. Mi -mi is the oscillation of f on the ith interval, so J-L(L1x) is the largest oscillation on any subinterval of the partition. If J-L(L1x) = Mi -mi, let c(L1x) and d(L1x) be the points in [xj-l, xi] where Mi and mj are attained. Existence of these points is guaranteed by the extreme value theorem 7.6, because f is continuous on the closed interval [xj_1,xj]· Then J-L(L1x) = f(c(L1x))x-f(d(L1x)), and lc(L1x)x-d(L1x)l :::; L1x. (i) Hence U!(f, L1x)x-L(f,L1x) L:=l(Mi(-mi)L1xi < L:=l J-L(L1x)L1xi J-L(L1x) 2:1 L1xi J-L(L1x)(bx-a), and so Ux(f, L1x)x-Lx(f, L1x) ::; f(c(L1x))x-f(d(L1x))x(bx-a). (ii) Thus we have shown that for all real L1x > 0 there exist c( L1x), d( L1x) E :a, b] such that (i) and (ii) hold. But then this transfers to *JR. Choosing .dx to be a positive infinitesimal, the transfer of (i) gives c(L1x) "'-' d(L1x)in *[a, b], so by taking their shadow we get a real r E [a, b] with c(L1x)x::::: rx::::: d(L1x). 108 9. The Riemann Integral Then by continuity of f, f(c(L1x)) f(r) f(d(L1x)), so f(c(L1x)) -f(d(L1x)) is infinitesimal (this is just a repetition of the proof from Theorem 7.7.2 that f is uniformly continuous on [a, b]). Since i (bx-a) is limited, transfer of (ii) then implies that Ux(f, L1x)x-Lx(f, L1x) is infinitesimal, as desired. D Exercise 9.1.2 (Monotonic Functions) Suppose that f is nondecreasing on [a, b] in the sense that f(x) < f(y)whenever xx:::; y. Show that for L1x E JR+, Ux(f, L1x)x-Lx(f, L1x) :::; L1x(f(b)x Similarly, ifnf is nonincreasing in the sense that f(x) > f(y) whenever x :::; y, show that U(f, L1x)x-Lx(f, L1x) :::; L1x(f(a)x-f(b)). A function is monotonic on [a, b] if it is either nondecreasing or else nonincreasing. Prove that Theorem 9.1.1 holds for monotonic functions as well as for continuous ones. 9.2 The Integral as the Shadow of Riemann Sums It will now be shown that any continuous or monotonic function on a closed interval in JR is Riemann integrable (note that any such function is bounded). Let L1x1 and L1x2 be positive infinitesimals, with associated lower sums L1, L2 and upper sums U11U2 for a continuous or monotonic function f on [a, b]. Then L1 :::; U1 and L2 :::; U2. But since real upper sums dominate all real lower sums, and this continues to be true in the hyperreal case by transfer, we also have L2 :::; U1 and L1 :::; U2. Thus the possible relationships are f(a)). or or the corresponding statements with the subscripts interchanged. But L1 U1 and L2 U2 by Theorem 9.1.1 or Exercise 9.1.2, so it follows that in any case 9.2 The Integral as the Shadow of Riemann Sums 109 Also, since the associated ordinary Riemann sums lie between their corresponding upper and lower sums, these are also infinitely close: Sx(f, Llxl) Sx(f, L1x2). Altogether then, the Riemann sums determined by arbitrary positive infinitesimals are all infinitely close to each other, and moreover are bounded above and below by the real numbers m(b(-a) and M(b(-a), so are all limited, and hence have the same shadow. Thus we may conveniently define I: f(x)dx = sh(S(f,xLlx)) (iii) for any positive infinitesimal Llx. Now we show that this definition fulfills the characterising conditions (1) and (2) of Section 9.1 for Riemann integrability. First observe that if P is any standard partition of [a, b], then taking an infinitesimal Llx yields Lx(f,xP)x::; Ux(f,xLlx) I: f(x)dx Lx(f,Llx)x::; U(f,P), and so as Lx(f, P) and Ux(f, P) are real, L(f,P)x::; I: f(x)dxx::; U(f,P). Secondly, given a positive c E lR then by Theorem 9.1.1 there exists a hyperreal Llx (namely any positive infinitesimal) such that Ux(f, Llx)x-Lx(f, L1x) < c:, and so by existential transfer this holds for some real Llx. This completes our proof using infinitesimals that a continuous or monotonic function f is Riemann integrable on [a, b], with integral defined as in (iii) . Notice that (iii) implies the standard characterisation of the integral as a limit: I: f(x)dx = lim.:1x->O+ S(f, L1x). Here S(f, L1x) was obtained formally as the extension of a standard function. In Section 12.7 we will see how to obtain it by a more explicit summation of terms f(x)L1x, with L1x infinitesimal, over a "hyperfinite" partition of [a, b]. If a function f is Riemann integrable on [a, b], then in the standard theory it is shown that the upper and lower sums approximate each other arbitrarily closely, by showing that for any given e E JR+ there exists a 6 E JR+ such that (VL1x E JR+) (L1x < 8 implies U(J, Llx)x-L(f, L1x) < c:). Transferring this and taking Llx to be infinitesimal, we get Ux(f, L1x) L (!,L1x) < c:. Since c is an arbitrary member of JR+ here, it follows that Ux(f, L1x) and Lx(f, L1x) are infinitely close. Hence 110 9. The Riemann Integral i Ux(f, Llx) Lx(J,Llx) for all positive infinitesimals Llx. But it was just this property that enabled us to obtain I: f(x)dx by the definition (iii). The property is therefore equivalent to Riemann integrability of a bounded function f on [a, b]. Exercise 9.2.1 ix For each (standard) n E N, let Ux(f, n), Lx(f,n), S(f, n) be the upper, lower, and ordinary Riemann sums for the partition determined by the number Llx = Prove that if n E *N is unlimited, then Lx(f, n) Sx(f, n) Ux(f, n). Show how the definition and proof of existence for the Riemann integral could be developed just using these functions of (hyper )natural numbers. 9.3 Standard Properties of the Integral Iff and g are integrable on [a, b] in JR., then • I: cf(x)dx = c I: f(x)dx. • I: f(x) + g(x)dx =I: f(x)dx +I: g(x)dx. • I: f(x)dx =•I: f(x)dx +I: f(x)dx if a :s; c :s; b. • I: f(x)dx :s; I: g(x)dx if f(x) :s; g(x) on [a, b]. • m(b(-a) :s; I: f(x)dx :s; M(b-a) if m :s; f(x) :s; M on [a, b]. Here is a concise proof via infinitesimals of the third (juxtaposition) property. If Llx = (cx-a)/n with n EN, then it is readily seen that Sx(J, Llx) = Sx(f, Llx) + Sx(J, Llx). (iv) Hence by transfer, this equation holds when Llx = (c -a)/N with N unlimited in *N. But then Llx is infinitesimal, so applying the shadow map to (iv) and invoking (iii) gives the result. Exercise 9.3.1 Derive proofs in this vein for the other properties of the integral listed above. 9.4 Differentiating the Area Function 111 9.4 Differentiating the Area Function Integration and differentiation are processes springing from quite different intuitive sources, but they are intimately related and, as every calculus student knows, are in a sense inverses of each other: differentiating the integral gives back the original integrand. This fundamental result is explained by examining the area function F, defined by :x F(x) = Jxf(t)dt for x E [a, b], where f is continuous on [a, b]. The key to the relationship between differentiation and integration is the following fact. : Theorem 9.4.1 The function F(x) = Jxj(t)dt is differentiable on [a, b), and its derivative is f. (This includes the right-and left-hand derivatives at the end points of the interval.) There is a very intuitive explanation of why this relationship should hold. The increment LlF = F(x + Llx)x-F(x) of F at x corresponding to a positive infinitesimal Llx is closely approximated by the area of the rectangle of height f(x) and width .dx, i.e, by f(x)Llx. Thus the quotient should be closely approximated by f(x)itself. Does "closely approximate" here mean f(x)? Well, observe that l1F is bounded above and below by f(x1)Llx and f(x2)Llx, where XI and x2 are points where f has its greatest and least values between x and x+ Llx, so lies between f(xi) and j(x2). But XI and x2 are infinitely close to x, hence by continuity j(x1) and j(x2) are infinitely close to j(x), and therefore so is . We will now use transfer to legalise this intuitive approximation argument. First, if Llx is a positive real number less than bx-x, then by juxtaposition of the integrals, +Llx :x F(x + Llx)x-F(x) = Jxf(t)dt. But on the interval [x, x + Llx], f attains maximum and minimum values at some points XI and x2, and so Hence F(x + Llx)x-F(x) f(X2) < < f(XI). (v) Llx 112 9. The Riemann Integral Thus for all real Llx E (0, b-x) there exist x1, x2 such that x ::;: x1,x2 ::;: x+Llx, and (v) is true. Hence by transfer, if L1x is any positive infinitesimal, then there are hyperreal X1,x2 E [x, x + L1x] for which (v) holds. But now x x + L1x, so x1 x "' x2, and hence by the continuity of f at x, f(xl) f(x) f(x2), from which (v) yields "'f( x). F(x + Llx)o-F(x) - L1x Similarly, this conclusion can be derived for any negative infinitesimal L1x. It follows (Theorem 8.1.1) that f(x) is the derivative ofnF at x, proving Theorem 9.4.1. Theorem 9.4.2 Fundamental Theorem of Calculus. If a function G has a continuous derivative f on [a, b), then J: f(x)dx = G(b)o-G(a). Proof This follows from Theorem 9.4.1 by standard arguments that require no ideas of limits or infinitesimals. For if F(x) = fax f(t)dt, then on [a, b) we have (G(x) -F(x))' = f(x) -f(x) = 0, so there is a constant c with G(x) -F(x) = c. This implies G(b) -G(a) = F(b) -F(a). But : F(b)o-F(a) = Jxf(t)dt. 0 9.5 Exercise on Average Function Values Let f be continuous on [a, b] JR. Define the "sample average" function Av by putting, for each n EN, f(xo) + + · Av(n) = · · n where Xio= a+ i(b-a)fn. Prove that if N E *N is unlimited, then 1 1b Av(N) b _a a f(x)dx (i.e., the average value ofnf on [a, b] is given by the shadow of Av(N)).