Point 2. Definite integral
Def. 2. Let a function be given on a segment a, b (fig. 2).
1. We divide the segment into n parts (subintervals)
by points (division points)
;
let
,
.
2.
We take an arbitrary point
in every subinterval
find the
Fig. 2
value of the function
at this point and multiply this value by the length
of the subinterval.
3. Adding all these products we get a sum (Cauchy1-Riemann2 integral sum)
. ( 8 )
4. If there exists the limit of the integral sum
(8) as
,
this limit is called the definite
integral of the function
over the segment a,
b
and is denoted
.
( 9 )
We read the left side of (9) as “definite
integral from a to
b of
”.
have
the same names as for indefinite integral; a
is called the lower limit of integration, b
the upper limit of integration.
Def. 3. A
function
is called integrable
on [over] the segment a,
b,
if its definite integral (9) exists.
Theorem 1 (existence theorem). If a function is continuous one on the segment a, b then it is integrable over this segment.
Geometric sense of
a
definite integral. If a function is
non-negative,
,
then by (2), (3) its definite integral is the area of a curvilinear
trapezium (1), fig. 1,
( 10 )
Economical sense of a definite integral. If a function is a labour producti-vity of some factory, then its produced quantity U during a time interval 0, T by vir-tue of (4), (5) is represented by a definite integral,
.
( 11 )
Physical sense of
a
definite integral. If a function
is the velocity of a material point then, on the base of (6), (7),
the length path L
traveled by the point during a time interval from t
= 0 to t =
T is given
by a definite integral
( 12 )
Ex. 1. Prove that
(
13 )
■The integrand
,
and so the integral sum (8) equals the length of the segment
,
that is
,
therefore
its limit, which is the integral (13), equals
.■
Note 1. Definite integral doesn’t depend on a variable of integration. It means that
( 14 )
Def. 4 (definite integral with equal limits of integration).
(
15 )
Def. 5 (interchanging limits of integration).
( 16 )
Point 3. Properties of a definite integral
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 c(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
.■