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1 (homogeneity). A constant factor k can by taken outside the integral sign,


■Integral sums for the left and right sides are equal, because of ,

therefore their limits are also equal.■

2 (additivity with respect to an integrand). If be two integrable functions then


Prove this property yourselves.

Corollary(linearity). For any two integrable functions and arbitrary constants


3 (additivity with respect to an interval of integration). For any a, b, c

if all three integrals exist.

■1) Let at first cÎ(a, b). We form an integral sum such that c be a division point. In this case (notations are clear)


and the passage to limit as gives the property in question.

2) Let now a disposition of points a, b, c is arbitrary, for example a < b < c. Using the first case and the definitions 4, 5 we’ll have


Integration of inequalities

4. If a < b and an integrand then


The integral is strictly positive if a function is continuous on the segment .

■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 functions on the segment .

■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 estimation 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 )

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


By Bolzano–Cauchy theorem for a function, continu-

Fig. 3 ous 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].

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