I ntegration of inequalities

4. If a < b and an integrand then

.

The integral is strictly positive if a function is continuous on the segment and if it doesn’t equal zero identically.

■Nonnegativity of the integral immediately follows from nonnegativity of the integral sum for the function Its strict positivity, as can be proved by more complicated reasonings, is the result of continuity of the function.■

5. If a < b and then

.

The integrals are connected by strict inequality in the case of continuity of the func-tions on the segment and if these functions don’t equal identically.

■It’s sufficient to apply the preceding property to a difference ■

Ex. 2. , because of on the segment .

6. If a < b then

( 17 )

■ It’s sufficient to apply the property 5 to the inequality

■

7 (two-sided estimate of a definite integral). Let a < b, and a function is continuous on a segment a, b. Then a double inequality is valid

. ( 18 )

■Proving follows from the property 5, the inequality on and the integral (13).■

Ex. 3. Estimate the integral .

and by the formula (18)

.

8. Mean-value theorem. If a function is continuous on a segment then there exists a point such that

( 19 )

Fig. 3 ■Let for example a < b. After dividing of both sides of the inequality (18) by positive number we get

.

By Bolzano³–Cauchy theorem for a function, continuous on a segment , there is a point such that

.

The case is studied by the same way. Do it yourselves.■

Geometric sense of mean-value theorem (fig. 3). The area of a curvilinear tra-pezium (1) equals the area of the rectangle ABCD with the same base AD=a, b and

the altitude .

Def 6. The expression

( 20 )

is called the mean value (the average value) of the function on the segment a, b.

Point 4. Definite integral as a function of its upper variable limit

Let xa, b. We consider a function

, ( 21 )

that is a definite integral with variable upper limit x. Geometrically (for ) this integral represents Fig. 4 the area of a part of a curvilinear trapezium

which lies between straight lines (fig. 4).

Theorem 2. If a function is continuous on a segment a, b then for any

the derivative of the integral (21) equals

, ( 22 )

that is the derivative of a definite integral with a variable upper limit x, with respect to this limit x, equals the value of the integrand at the point x.

■By definition of the derivative

.

Using the additivity of a definite integral with respect to an interval of integration we get

Let for example . By virtue of the mean-value theorem there is a point in the interval such that

.

This point as . Taking into account continuity of the function f we get

.■

Corollary (Fundamental theorem of the integral calculus). Every function which is continuous one on a segment a, b has a primitive on a, b.

■One of such primitives is a definite integral (21) with variable upper limit x.■

Ex. 4. The derivative of the function equals