The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 127 Friday, March 12, 2004 12:39 PM

Principles of Calculus 127

INTEGRATION

Differentiation addresses the problem of defining the instantaneous rate

of change, whereas integration addresses the problem of calculating the

area of an arbitrary figure. Areas are easily defined for rectangles and

triangles, and any plane figure that can be decomposed into these

objects. While formulas for computing the area of polygons have been

known since antiquity, a general solution of the problem was arrived at

first in the seventeenth century, with the development of calculus.

Riemann Integrals

Let’s begin by defining the integral in the sense of Riemann, so called after

the German mathematician Bernhard Riemann who introduced it. Con-

sider a bounded function y = f(x) defined in some domain which includes

the interval [a,b]. Consider the partition of the interval [a,b] into n disjoint

subintervals a = x 0 < x 1 < ... < xn–1 < xn = b, and form the sums:

n

SU n = ∑fM()xi(xi – xi – 1 )

i = 1

where fM()xi = supfx()x ∈ , [xi – 1 ,xi] and

n

SL n = ∑fm()xi(xi – xi – 1 )

i = 1

where fm ()xi = inf fx()x ∈ , [xi – 1 ,xi].

Exhibit 4.8 illustrates this construction. Sn U ,SLn are called, respec-

tively, the upper Riemann sum and lower Riemann sum. Clearly an infi-

nite number of different sums, SU L

n ,Sn can be formed depending on the

choice of the partition. Intuitively, each of these sums approximates the

area below the curve y = f(x), the upper sums from above, the lower

sums from below. Generally speaking, the more refined the partition the

more accurate the approximation.

Consider the sets of all the possible sums { } SU and {}L for every

n Sn

possible partition. If the supremum of the set {}SL (which in general

n

will not be a maximum) and the infimum of the set { } SU (which in gen-

n

eral will not be a minimum) exist, respectively, and if the minimum and

the supremum coincide, the function f is said to be “Riemann integrable

in the interval (a,b).”

If the function f is Riemann integrable in [a,b], then