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Riemann Integral

The length of an interval [a, b] on the real line is defined to be b-a. Let P : a = x0 < x1 < … < xk-1 < xk < … < xp-1 < xp = b denote a partitioning of the interval [a, b] in to subintervals Sk = [xk-1, xk], 1 £ k £ p. Let xk Î Sk be arbitrary and let Dk = xk-xk-1 denote the length of the interval Sk. Let f(x) be a bounded real valued function defined on the interval [a, b]. The sum

S(P) = S1£ k£ p f(xk)Dk,

is called a Riemann sum of f(x) associated with the partition P. Let mk denotes the inf of f(x) on Sk, i.e., the greatest lower bound of f(x) on Sk, and, Mk denotes sup of f(x) on Sk, i.e., the least upper bound of f(x) on Sk. The lower Riemann sum and the upper Riemann sum of f(x) for the partition P are defined, respectively, as

L(P) = S1£ k£ p mkDk, U(P) = S1£ k£ p MkDk,

where mk denotes the inf of f(x) on Sk and Mk denotes sup of f(x) on Sk. Clearly,

L(P) £ U(P).

The lower Riemann integral L (f), and the upper Riemann integral U (f) are defined by

L (f) = sup {L(P)}, U (f) = inf {U(P)},

the sup and the inf running over all partitions P of [a, b]. Note that for all partitions P: L(P) £ L (f), U (f) £ U(P).

Let Q be another partition. The partition R = PÈQ is obtained by taking its subintervals to be the intersections of subintervals of P with those of Q. Thus each subinterval in R is a subset of some subinterval in P as well as some subinterval in Q. Such an R is called a refinement of P. Since inf over a larger set is not larger, and the sup over a larger set is not smaller, it is clear that

L(P) £ L(PÈQ) £ U(PÈQ) £ U(Q).

Taking sup over P’s, L (f) £ U(Q). Next taking inf over Q’s, we get

L(f) £ U(f).

Hence, for any partition P,

L(P) £ L(f) £ U(f) £ U(P).

The function f(x) is called Riemann integrable over [a, b] if L (f) = U (f), and then this common value is defined to be the Riemann integral or the definite integral of f(x) written as ò(a,b) f(x)dx. The Riemann criterion for the Riemann integrability is:

Theorem. f(x) is Riemann integrable iff for each e > 0, there exists a partition P such that

U(P) - L(P) < e .

Proof: The necessity follows from the definition of sup and inf. For sufficiency, 0 £ U (f) - L (f) £ U(P) – L(P) < e . Since e > 0 is arbitrary, by letting it to tend to zero it follows that U (f) = L (f). #

For a partition P : a = x0 < … < xk-1 < xk < … < xp-1 < xp = b, the partition size is defined by D(P) = max1£ k£ p D k.

Theorem. Let f(x) be Riemann integrable on [a, b], and let {S(Pi)} be a sequence of Riemann sums with limi® ¥ D(Pi) = 0. Then limi®¥ S(Pi) = ò(a,b) f(x)dx.

Proof: Let e > 0 be arbitrary. By the Riemann criterion, there exists a partition P such that U(P)–L(P) < e . Let Qi be the union of the partitions Pi and P. Let m = inf f(x) and M = sup f(x). Then L(P) £ L(Qi) £ U(Qi) £ U(P), U(Pi)-U(Qi) £ pD(Pi)(M-m), and L(Qi)–L(Pi) £ pD(Pi)(M-m). Hence, U(Pi)-L(Pi) £ U(Qi)–L(Qi)+2pD(Pi)(M-m). It follows that, limsupn® ¥ [U(Pi)-L(Pi)] £ U(Qi)–L(Qi) £ U(P)–L(P) < e . Since e > 0 is arbitrary, it follows that limn®¥ [U(Pi)-L(Pi)] = 0. Hence, limi®¥ U(Pi) = ò(a,b) f(x)dx] = limi®¥ [L(Pi), so that by the sandwitch theorem, also, limi®¥ S(Pi) = ò(a,b) f(x)dx. #

Two useful classes of Riemann integrable functions are as follows. Define the partition size D(Pn) = max1£ k£ n Dk.

Theorem. If f(x) is continuous on [a, b], f(x) is Riemann integrable on [a, b].

Proof: Since f(x) is uniformly continuous on [a, b] given any e > 0, there exists a d > 0 such that |f(x)-f(y)| < e , for all |x-y| < d . Hence for any partition P such that D(P) < d , U(P) – L(P) < e(b-a), and the Riemann criterion is satisfied. Hece f(x) is Riemann integrable on [a, b]. #

Theorem. If f(x) is monotone on [a, b], f(x) is Riemann integrable on [a, b].

Proof: For definiteness we assume that f(x) is non-decreasing, i.e., x £ y implies f(x) £ f(y). Then, for any partition P, U(P)-L(P) = S 1£ k£ p-1 [f(xk)-f(xk-1)]Dk £ D(P)[f(b)-f(a)] and so taking D(P) sufficiently small the Riemann criterion is satisfied. #

Area Under a Curve

The area of a rectangle [a, b]´[c, d] is defined to be the product of its sides, namely, (b-a)´(d-c). The area under a curve C : y = f(x) ³ 0, a £ x £ b, where y(x) is a continuous function on [a, b] means the area of the region R bounded by C, x-axis and the vertical lines x = a, and x = b. To motivate the definition of this area, consider a partition Pn of [a, b]. Since, È1£k£n [xk-1, xk]´[0, mk] Ì R Ì È1£k£n [xk-1, xk]´[0, Mk] and the areas of the unions of rectangles are respectively L(Pn) and U(Pn) we should have L(Pn) £ A £ U(Pn) for all Pn. Hence, as U(Pn) and L(Pn) both converge to ò(a,b) f(x)dx, we arrive at: A = ò(a,b) f(x)dx. Hence for a region defined by R12 = {(x, y) : y1(x) £ y £ y2(x), a £ x £ b} the area becomes: A = ò(a,b) [y2(x)- y1(x)]dx. The area of a region that can be divided in to a finite numbers of sub-regions of type R12 can thus be obtained as the sum of the areas of the individual sub-regions. For more general regions the areas could be found by a limiting process involving series and improper integrals.

Multiple Integral

A subset S of the Euclidean n-space Rn = {x = (x1, x2, … , xn)¢ : xi Î R, 1 £ i £ p} is called bounded if it is contained inside a sphere {x : x12+ x22+…+ xn2 = R2}of a positive radius R centered at the origin. S is called open if it contains a ball B(z, r) ={x : |x-z|2 = (x1-z1)2 + (x2-z2)2 +…+ (xn-zn)2 < r2} of a positive radius r about each of its points z. Given a set D in R n, its boundary, written bd D or D, consists of points z in R n such that B(z, r) has a non-empty intersection with D, as well as R n\D, for each r > 0. D is callled closed if bd D Ì D. A bounded closed set in R n will be called a region or a domain in R n if its boundary consists of a finite number of (n-1)-dimensional piecewise smooth orientable (i.e., possessing a continuously differentiable unit normal vector at each of its points) surfaces.

Given a domain D in R n, with its boundary consisting of a finite number of (n-1)-D surfaces, to define the Riemann integral of a function f(x) on D, consider a partitioning P of D into small n-D volume elements Dk of known n-D volumes Dk, 1 £ k £ p. A Riemann sum associated with P is given by

S(P) = S1£ k£ n f(xk)Dk,

where x k is some point in the k-th partition element Dk. The lower and upper Riemann sums associated with P are

L(P) = S 1£ k£ p mkDk, U(P) = S 1£ k£ p MkDk.

Clearly:

L(P) £ U(P).

The lower Riemann integral L (f), and the upper Riemann integral U (f) are defined by

L(f) = sup {L(P)}, U(f) = inf {U(P)},

the sup and the inf running over all partitions P of D. Note that for all partitions P: L(P) £ L(f), U(f) £ U(P).

Let Q be another partition. The partition R = PÈQ is obtained by taking its volume elements to be the intersections of volume elements of P with those of Q. Thus each volume element in R is a subset of some volume element in P as well as some volume element in Q. Such an R is called a refinement of P. Since inf over a larger set is not larger, and the sup over a larger set is not smaller, it is clear that

L(P) £ L(PÈQ) £ U(PÈQ) £ U(Q).

Taking sup over P’s, L(f) £ U(Q). Next taking inf over Q’s, we get

L(f) £ U(f).

Hence, for any partition P,

L(P) £ L(f) £ U(f) £ U(P).

The function f(x) is called Riemann integrable over D if L(f) = U(f), and then this common value is defined to be the Riemann integral or the multiple integral of f(x) written as òD f(x)dx, or ò … òD f(x1, x2, … , xn)dx1dx2 … dxn. In particulr a double integral is denoted by òòD f(x, y)dxdy, and a triple one by òòòD f(x, y, z)dxdydz.The Riemann criterion for the Riemann integrability is:

Theorem. f(x) is Riemann integrable on a region D in Rn iff for each e > 0, there exists a partition P of D such that: U(P) - L(P) < e .

Proof: The necessity follows from the definition of sup and inf. For sufficiency, 0 £ U(f) - L(f) £ U(P) – L(P) < e . Since e > 0 is arbitrary, by letting it to tend to zero it follows that U(f) = L(f). #

For a partition P : Èk Dk, the partition size is defined to be the smallest number D(P) such that each of Dk could be enclosed within a sphere of diameter D(P).

Theorem. Let f(x) be Riemann integrable on a region D in Rn and let {S(Pi)} be a sequence of Riemann sums with limi® ¥ D(Pi) = 0. Then limi® ¥ S(Pi) = òD f(x)dx.

Proof: Let e > be arbitrary. By the Riemann criterion, there exists a partition P such that U(P)–L(P) < e . Let Qi be the union of the partitions Pi and P. Let m = inf f(x), M = sup f(x) and DV(P) denote the volume of a sphere of diameter D(P). Then L(P) £ L(Qi) £ U(Qi) £ U(P), U(Pi)-U(Qi) £ pDV(Pi)(M-m), and L(Qi)–L(Pi) £ pDV(Pi)(M-m). Hence, U(Pi)-L(Pi) £ U(Qi)–L(Qi)+2pDV(Pi)(M-m). It follows that, limsupi®¥ [U(Pi)-L(Pi)] £ U(Qi)–L(Qi) £ U(P)–L(P) < e . Since e > 0 is arbitrary, it follows that limn®¥ [U(Pi)-L(Pi)] = 0. Hence, limi®¥ U(Pi) = òD f(x)dx = limi®¥ L(Pi), so that by the sandwitch theorem, also, limi®¥ S(Pi) = òD f(x)dx. #

Theorem. If f(x) is continuous on a region D in Rn, f(x) is Riemann integrable on D.

Proof: Since f(x) is uniformly continuous on D given any e > 0, there exists a d > 0 such that |f(x)-f(y)| < e , for all |x-y| < d . Hence, if VD denotes the n-D volume of D, for any partition P such that D(P) < d , U(P)–L(P) < eVD, and the Riemann criterion is satisfied. Hece f(x) is Riemann integrable on D. #

An upper bound for the error in approximation by a Riemann sum

Theorem. Let D be a bounded closed domain in R n and f(x) a continuous function on D. Let w(f; d ) = sup|x-y|£ d |f(x)-f(y)|. Then, if VD denotes the volume of D, for any partition P: |òD f(x)dx - S(P)| £ w(f; D(P))VD.

Proof: Let Pi be a sequence of refinements of P with D(Pi) ® 0. Then, |S(Pi)-S(P)| £ w(f; D(P))VD. Taking limit as i ® ¥ , we have: |òD f(x)dx - S(P)| £ w (f; D(P))VD. #

Theorem (Rectangular Partitions). Let f(x) be Riemann integrable on a region D in Rn. If the boundary of D consists of a finite number of piecewise smooth orientable (n-1)-D surfaces and P is a partition of D obtained by drawing hyperplanes perpendicular to coordinate axes and S(P) is a Riemann sum associated with the largest number of rectangular paralellopipeds that are contained in D, or the smallest number of rectangular paralellopipeds that contain D, then if f(x) is continuous on D, limD(P)®0 S(P) = òD f(x)dx.

Proof: For the parallelopipeds from within D the volume of the left out region does not exceed D(D)AD, where AD denotes the (n-1)-D surface area of D. Hence the full Riemann sum is missed by a quantity not exceeding MD(D)AD ® 0,as D(D) ® 0, where M = maxxÎD f(x). In the case of parallelopipeds from without, similarly, the Riemann sum is exceeded by a quantity not exceeding MD(D)AD. The result, then, follows from the Riemann integrability of F(x). #

Fubini's Theorem (for functions of two variables). Let D = {(x, y) : g(x) £ y £ h(x), a £ x £ b} be a bounded region in the plane, where g(x) and h(x) are continuous functions on [a, b]. If f(x, y) is a continuous function on D, then: òòD f(x, y)dxdy = ò(a,b) [ò(g(x),h(x)) f(x, y)dy]dx.

Proof: By the uniform continuity of continuous functions on bounded closed sets, given an arbitrary e > 0, there exist h, k > 0 such that |f(x, y)-f(z, t)| < e , whenever |x-z| < h, |y-t| < k. Consider a Riemann sum associated with a rectangular subdivision from within of type Si Sj f(xi, yj)DxDy with 0 < Dx < h, 0 < Dy < k. Let M = maxD |f(x,y)| and let DLi denote the sum of the lengths of the curves y = g(x) and y = h(x) on the i-th subinterval on the x-axis. Then, | Sj f(xi, yj)Dy - ò(g(x),h(x)) f(xi, y)dy| < MDLi + e [h(x)-g(x)], so that |Si [Sj f(xi, yj)Dy - [ò(g(x),h(x)) f(xi, y)dy]Dx| £ M(Lg+Lh)Dx + e(b-a) max[a,b] [h(x)-g(x)]. Also, |Si ò(g(x),h(x)) f(xi, y)dy]Dx - ò(a,b) [ò(g(x),h(x)) f(x, y)dy]dx| < (b-a)e max[a,b] [h(x)-g(x)]. Since e > 0 is arbitrary, it follows from the Riemann integrability that

òòD f(xy)dxdy = lim(Dx,Dy)®(0,0) Si Sj f(xi, yj)DxDy = ò(a,b) [ò(g(x),h(x)) f(x, y)dy]dx. #

By the symmetry of the definition of òòD f(xy)dxdy with respect to the coordinate axes, we immediately have:

Corollary. Let D = {(x, y) : p(y) £ x £ q(y), c £ y £ d}, where p(y) and q(y) are piecewise smooth functions on [c, d]. If f(x, y) is a continuous function on D, then: òòD f(xy)dxdy = ò(c,d) [ò(p(y),q(y)) f(x, y)dx]dy.

Corollary. If D = {(x, y) : a £ x £ b, c £ y £ d} and f(x, y) is a continuous function on D,

ò(a,b) [ò(c,d) f(x, y)dy]dx = òòD f(x, y)dxdy = ò(c,d) [ò(a,b) f(x, y)dx]dy.

Fubini's Theorem. Let D be a bounded region in Rn described by: D = {x Î Rn : fi(xi+1, … , xn) £ xi £ gi(xi+1, … , xn), 1 £ i £ n-1, a £ xn £ b}, where the (n-i)-dimensional surfaces xi = fi(xi+1, … , xn), xi = gi(xi+1, … , xn), 1 £ i £ n-1 are piecewise smooth. If f(x) is continuous on D, then

òD f(x)dx º ò … òD f(x1, x2, … , xn)dx1dx2 … dxn = ò(a,b) [ò(fn-1, gn-1) … [ò (f1, g1) f(x1, x2, … , xn)dx1] … dxn-1]dxn.

Proof: Choosing a rectangular partition, equispaced function values f(x k)'s and re-arranging the Riemann sum first with respect to x1, then x2, and so on and lastly with respect to xn and recognizing the individual sums as Riemann sums for definite integrals, the result follows from the Riemann integrability of continuous functions, along lines similar to those for the case of two variables. #

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