MATHEMATICS  THEORY



This section introduces the definition of the integral
and its properties. 





Definite Integral



In
Area section,
the limit was used to approximate
the area between a curve and xaxis. This type of
limit often occurs in engineering research,
such as finding the fluid pressure over an area or the work used in a
process. A special name and notation is
given to this kind of limit  definite integral. It states: 



Definite Integral 

If a function f is defined
on a closed interval between a and b, and it is equally
divided into n
parts. So the length of each part is Δx
= (ba)/n, then the definite integral of function f from a to b is
(1)
In this equation, ∫ is the integration sign; a is the lower limit;
b is the upper limit; f(x) is the integrand. The procedure
for calculating an integral is called integration and is
called the Riemann sum. The definite integral is sometimes called the Riemann
integral. 





The definite integral of the function f(x), ,
is a number. When the function is positive, an integral can be interpreted
as an area.
=
the area under the graph of function from a to b 



Integral and area
Integral and area 

In general, a definite integral can be interpreted as
the difference of areas:
in which A_{1} is the sum area of the region above the xaxis
and A_{2} is
the sum area of the region below the xaxis as shown on the left.
For example, the integral function f(x) = x between
2 and 4 can be calculated as the difference of the the two triangles.
Although this problem can be calculated, some of the integral
can't be calculated. 





Integrable

Integrable Discontinuous Function


It is clear that, if exist, then the function f(x) is integrable. The prerequisite for existing is the function f(x) must be continuous or monotonic. Therefore, if a function is either continuous or monotonic on a close interval between a and b, then the function is integrable on this interval. If the function is discontinuous, its integral might exist, such as the step function shown on the left.
Any function that is integrable
between a and b can be expressed as






Midpoint Rule



In order to calculate the limit, the right
endpoints are generally used. If the midpoint is used, the limit will
be an approximate
value and the integral will be
where Δx = (b  a)/n
and
To assist in evaluating integrals, some basic properties
of integrals are list below. 





Properties of the Integral



If the integral of a function exist in
[a, b], then 



where c is any constant

where c is any constant


 if
f(x) ≥0.
 if f(x) ≥g(x).

if m ≤ f(x) ≤ M .



