The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 132 Friday, March 12, 2004 12:39 PM

132 The Mathematics of Financial Modeling and Investment Management

as the left limit

b x

∫ fx()xd = lim fu()ud

a x →b – ∫a

A similar definition can be established for the lower integration

limit. Improper integrals exist only if these limits exist. For instance, the

integral

(^11 11)  1 

∫ --- xd = lim – ------ = lim ------–^1  = ∞

0 x x → 0 + x^2 x → 0 +^2 

0 x

does not exist.

THE FUNDAMENTAL THEOREM OF CALCULUS

The fundamental theorem of calculus shows that integration is the

inverse operation of derivation; it states that, given a continuous func-

tion f, any of its indefinite integrals F is a differentiable function and the

following relationship holds:

x

df u()ud

dF x() ∫ a ()

--------------- = --------------------------- = fx

dx dx

If the function f is not continuous, then the fundamental theorem

still holds, but in any point of discontinuity the derivative has to be

replaced with the left or right derivative dependent on whether or not

the function f is left or right continuous at that point.

Given a continuous function f, any function F such that

dF x() ()

--------------- = fx

dx

is called a primitive or an indefinite integral of the function f. It can be

demonstrated that any two primitives of a function f differ only by a

constant. Any primitive of a function f can therefore be represented

generically as an indefinite integral plus a constant.