Mathematical Fundamentals
131
defined integral
indefinite integral
A.7.2
Integrals of Functions
For the purpose of this book, it is sufficient to informally describe the defined
integral of a function f (x), x R as the area delimited by the function curve
and the horizontal axis in the interval between two edges a e b (see fig. 10).
When the curve stays below the axis the area has to be considered negative,
and positive when it stays above the axis. The defined integral is represented in
compact notation as
b
a
f (x)dx ,
(42)
and it takes real values.
y
x
a
b
0
y=f(x)
Figure 10: Integral defined as an area
In order to compute an integral we can use a limiting procedure, by approxi-
mating the curve with horizontal segments and computing an approximation of
the integral as the sum of areas of rectangles. If the segment width approaches
zero, the computed integral converges to the actual measure.
There is a symbolic approach to integration, which is closely related to func-
tion derivation. First of all, we observe that for the integrals the properties of
linear operators do hold:
· The integral of a sum of functions is the sum of integrals of the single
functions
· The integral of a product of a function by a constant is the product of the
constant by the integral of the function.
Then, we generalize the integral operator in such a way that it doesn't give a
single number but a whole function. In order to do that, the first integration
edge is kept fixed, and the second one is left free on the x axis. This newly
defined operator is called indefinite integral and is indicated with
F (x) =
x
a
f (u)du .
(43)
The argument of function f (), also called integration variable, has been called
u to distinguish it from the argument of the integral function F ().
The genial intuition, that came to Newton and Leibniz in the XVII century
and that opened the way to a great deal of modern mathematics and science,
was that derivative and integral are reciprocal operations and, therefore, they
are reversible. This idea is translated in a remarkably simple formula:
F (x) = f (x) ,
(44)