Complex Analysis - Maths KU

5.1 Real Integrals 237
3. Let �P � be the norm of the partition P of [a, b], that is, the length
of the longest subinterval.
4. Choose a number x ∗ k in each subinterval [xk−1, xk] of[a, b]. See
Figure 5.1.
5. Formn products f(x ∗ k )∆xk, k =1,2,... , n, and then sumthese
products:
n�
f(x ∗ k)∆xk.
k=1
Definition 5.1 Definite Integral
The definite integral of f on [a, b] is
� b
a
f(x) dx = lim
�P �→0
n�
f(x ∗ k)∆xk. (2)
Whenever the limit in (2) exists we say that f is integrable on the
interval [a, b] or that the definite integral of f exists. It can be proved that
if f is continuous on [a, b], then the integral defined in (2) exists.
The notion of the definite integral � b
f(x) dx, that is, integration of a real
a
function f(x) over an interval on the x-axis from x = a to x = b can be
generalized to integration of a real multivariable function G(x, y) on a curve
C frompoint A to point B in the Cartesian plane. To this end we need to
introduce some terminology about curves.
Terminology Suppose a curve C in the plane is parametrized by a set
of equations x = x(t), y = y(t), a ≤ t ≤ b, where x(t) and y(t) are continuous
real functions. Let the initial and terminal points of C, that is, (x(a), y(a))
and (x(b), y(b)), be denoted by the symbols A and B, respectively. We say
that:
(i) C is a smooth curve if x ′ and y ′ are continuous on the closed interval
[a, b] and not simultaneously zero on the open interval (a, b).
(ii) C is a piecewise smooth curve if it consists of a finite number of
smooth curves C1, C2, ...,Cn joined end to end, that is, the terminal
point of one curve Ck coinciding with the initial point of the next curve
Ck+1.
(iii) C is a simple curve if the curve C does not cross itself except possibly
at t = a and t = b.
(iv) C is a closed curve if A = B.
(v) C is a simple closed curve if the curve C does not cross itself and
A = B; that is, C is simple and closed.
k=1