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 x-axis from a to b in opposite direction, each difference, x1 - ax2 - x1,  . . .  ,  b - xn - 1 of Riemann sums becomes negative while function values, f (xi') can stay unchanged, therefore
The definite integral changes sign if the limits of integration interchange.
3)  The constant multiple rule
Suppose  f (xis integrable over the interval [a, b]  and c is any real number, then  c f (xis 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 y-axis, 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) < 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
right-hand 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 a1 instead of a, such that a1 > a then
Thus, as the point a1 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 F0 that correspond to any chosen area over the interval [a, a1], as shows the above figure.
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