The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 129 Friday, March 12, 2004 12:39 PM

Principles of Calculus 129

I = lim S

max∆xi → 0 n

In the above, the limit operation has to be defined as the limit for

any sequence of sums Sn as for each n there are infinitely many sums.

Note that the function f need not be continuous to be integrable. It

might, for instance, make a finite number of jumps. However every

function that is integrable must be of bounded variation.

Properties of Riemann Integrals

Let’s now introduce a number of properties of the integrals (we will

state these without proof). These properties are simple mechanical rules

that apply provided that all integrals exist. Suppose that a,b,c are fixed

real numbers, that f,g,h are functions defined in the same domain, and

that they are all integrable on the same interval (a,b). The following

properties apply:

Properties of Riemann Integrals

a

Property 1 ∫ fx()xd = 0

a

c b c

()xd = ∫

a

fx

b

Property 2 ∫ fx ()xd + ∫ fx()xd , abc≤≤

a

b b b

Property 3 hx() α= fx() β+ gx()⇒ ∫ hx()xd = α ∫ fx()xd + β ∫ gx()xd

a a a

b b b

Property 4 ∫ f′()xgx()xd = fx()gx()a – ∫ fx()g′()x xd

a a

■ Properties 1 and 2 establish that integrals are additive with respect to

integration limits.

■ Property 3 is the statement of the linearity of the operation of integra-

tion.

■ Property 4 is the rule of integration by parts.

Now consider a composite function: h(x) = f(g(x)). Provided that g is

integrable on the interval (a,b) and that f is integrable on the interval corre-

sponding to all the points s = g(x), the following rule, known as the chain

rule of integration, applies: