Integral calculus
      The definite and indefinite integrals
      The area between the graph of a function and the x-axis over a closed interval
         Geometric interpretation of the definite integral
      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
The area between the graph of a function and the x-axis over a closed interval
Geometric interpretation of the definite integral
Until now we assume that the integrand, the function that is integrated, to be nonnegative or  f(x) > 0 for all x in an interval [a, b].
Now suppose f(x) < 0 in the whole interval or in some of its parts then, the areas of regions between the graph of  f and the x-axis, which lie below or above the x-axis, differ in the sign of  f(x).
 
Therefore, the definite integral yields the algebraic sum of these areas taking regions below the x-axis negative, as show the figures above.
So, if the graph of  f looks as in the left figure above then
hence the definite integral represents the algebraic sum of the areas above and below the x-axis.
Thus, as the right figure above shows, follows that
as the area A2, lying under the arc of the sinusoid in the interval [p, 2p] is congruent to the area A1 in [0, p] but with opposite sign.
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(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 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.
Calculus contents E
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