
Integral
calculus 

The
definite and indefinite integrals 
Properties of the definite integral 
The definite integral
over interval of zero length 
Reverse
order of integration 
The constant multiple rule 
The integral of the sum or difference of two functions 
Internal addition of the definite integral 
The definite integral of an odd function 
The definite integral of an even function 
The definite integral of a nonnegative and nonpositive functions 
Relationship of the definite integrals of two functions over the same
interval of integration 
Absolute integrability 
The mean value theorem 
The average value of a function over the given interval 






Properties
of the definite integral 
1)
The definite integral
over interval of zero length 
Since
the definite integral we evaluate as the limit of Riemann sums, the
basic properties of limits hold for integrals
as well. 
Thus, the limit of Riemann sums show
the first property


as
the upper limit b
tends to lower limit a,
written b
®
a. 

2)
Reverse
order of integration 
By
reversing the upper and lower limits of integration to b
< a, that is, passing
through the xaxis
from a
to b in
opposite direction, each difference, x_{1}

a, x_{2} 
x_{1}, . . . , b 
x_{n}_{
 1}
of Riemann sums becomes negative
while function values, f
(x_{i}')
can stay unchanged, therefore 

The definite integral
changes sign if the limits of integration interchange. 

3)
The constant multiple rule 
Suppose
f
(x) is integrable over the
interval [a,
b] and c
is any real number, then c
f
(x) is integrable over
[a,
b] such that 

If
integrand is multiplied by a constant, we can factor out the constant. 

4)
The integral of the sum or difference of two functions 
Suppose
f(x)
and g(x)
are integrable over the
interval [a,
b] then 

The
integral of the sum or difference of two functions is the sum or
difference of the integrals of each function 

5)
Internal addition of the definite integral 
Suppose
f
(x)
is integrable over the
interval [a,
b] and c
is a point inside the interval, i.e., a
< c < b then, the
additive
property of the definite integral holds 

Let
for example c
lies outside the interval such that c
< a < b, and
assume f
(x)
is integrable over the
interval [c,
b] then, 

Therefore,
the above rule holds for any arrangement of three points, a,
b
and c. 

6)
The definite integral of an odd function 
If
the integrand is an odd function f
(x) = 
f
(x),
the graph of which is symmetrical about the origin, with the
interval of integration [
a,
a],
then 


7)
The definite integral of an even function 
If
the integrand is an even function f
(x) = 
f
(x), the graph of
which is symmetrical about the yaxis,
with the
interval of integration [
a,
a],
then 


8)
The definite integral of a nonnegative and nonpositive functions 
Suppose
f
(x)
is integrable over the
interval [a,
b] and f
(x) >
0 but not identically
equal to zero, then 

If
f
(x)
is integrable over the
interval [a,
b] and f
(x) <
0 but not identically
equal to zero, then 


9)
Relationship of the definite integrals of two functions over the same
interval of integration 
Suppose
f
(x) and g
(x) are integrable over the
interval [a,
b] and f
(x) >
g
(x) for all x
in [a,
b], then 


10)
Absolute integrability 
Suppose
f
is integrable over the
interval [a,
b], it can be shown
that  f
(x) is also
integrable on [a,
b] 

Since,


f
(x) <
f
(x) <

f
(x), then 

what
means the same as the above. 

11)
The mean value theorem 
Suppose f
(x)
is integrable over the
interval [a,
b] and m
and M
are minimal and maximal value of the function,
that is m
<
f
(x) <
M
for all x
in [a,
b], then 

Geometric
meaning of the above inequality is that the area under the graph of f
(x) over the
interval [a,
b] is contained inside
the rectangles with the same base (b

a)
and of the heights m
and M. 
Since f
(x)
is continuous in the
interval [a,
b] it 
takes at least
one time each value between m
and M 
inside the interval. 
Therefore, there is at least one
value x
inside the 
interval [a,
b] such that 

meaning, there
exists the rectangle with the base 



(b

a)
and a height f
(x)
whose area equals the area under the graph of f
(x) over the
interval [a,
b]. 

12)
The average value of a function over the given interval 
For
a continuous function f
over an
interval [a,
b], the average value
of f
(x) is defined as 

Thus,
the average value of a function f
(x) over an
interval [a,
b] is equal to some
value of the function between
its minimal and maximal value inside the interval, as shows the above
figure. 

The
indefinite
integral 
Assume
the lower limit a
of an integral is fixed and the upper limit is changing. Then, starting
from the fixed
ordinate at a,
also changes the area A
under the curve as its endpoint ordinate moves parallel. 
As
we evaluate the integral over an interval whose 
righthand endpoint x
changes, the actual value of the 
area is
a function of x
usually denoted F(x),
so that 

The integral F(x)
is called the indefinite integral of f(x). 
Note
that the variable of integration is replaced by the 
letter t
as x now denotes the upper limit. 



Note also that, the value of the above integral will change whether
lower or upper limit of integration changes. 
So for example, we choose other fixed point a_{1}
instead of a,
such that
a_{1}
> a then 


Thus, as the point a_{1}
can be placed anywhere inside the interval [a,
x], the same function
f (x)
has infinite number
of indefinite integrals that differ only in a constant value F_{0}
that correspond to any chosen area
over the
interval [a,
a_{1}],
as shows the above figure. 









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