Definite and indefinite integrals, Area between graph of function and x-axis over closed interval, Geometric interpretation of definite integral, Relationship of definite integrals of two functions over same interval of integration

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 - a, x₂ - x₁, . . . , 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 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) < 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.

Calculus contents E