Lecture no. 21. Definite integral
POINT 1. PROBLEMS LEADING TO THE NOTION OF A DEFINITE INTEGRAL
POINT 2. DEFINITE INTEGRAL
POINT 3. PROPERTIES OF A DEFINITE INTEGRAL
POINT 4. DEFINITE INTEGRAL AS A FUNCTION OF ITS UPPER VARIABLE LIMIT
POINT 5. NEWTON-LEIBNIZ FORMULA
POINT 6. MAIN METHODS OF EVALUATION A DEFINITE INTEGRAL
Point 1. Problems leading to the concept ofa definite integral
P
roblem
1. Area of a curvilinear trapezium
Def. 1. Curvilinear
trapezium on the
-plane
is called a figure bounded by two straight lines
,
,
-axis
and a curve
(fig.1). It’s useful to denote a curvilinear trapezium as the next
point set on the
-
Fig. 1 plane
.
( 1 )
To define the notion of the area of a curvilinear trapezium (1), fig.1, we carry out the next construction.
1. With the help of points
we divide the segment into n parts (subintervals)
of the lengths
,
and
let
.
2. We take an arbitrary point
in every part
find the value of the function at this point and multiply it by
.
3. Adding all these products
we get a sum
( 2 )
-
the area of a step-type figure generated by rectangles with bases
and altitudes
.
4. Let tends to zero. If there exists the limit of the sum (2), it is called the area of the curvilinear trapezium (1) (fig.1) and is denoted
.
( 3 )
Problem 2. Produced quantity
Let
is a labour productivity of some factory at a time moment t.
Find its produced quantity
during a time interval
.
If
,
then
.
But as a rule
,
and we do as follows.
1. We divide the time segment 0, T into n parts
,
and put
.
2. We take an arbitrary point
in every part
find the value of the function
at this point and multiply it by
.
3. Adding all the products
we find an approximate value of the
produced quantity
during 0,
T,
that is
.
( 4 )
4. Tending
to zero we find the exact value of the produced quantity
.
( 5 )
Problem 3. Length path.
Find the length path L
traveled by a material point with a given velocity
during a time interval T
(from the time moment t
= 0).
If
then
.
For a variable velocity we do by the same way as in preceding problems.
1. We divide the interval 0, T into n parts
and put
.
2. In every time interval we take arbitrary moment , find the value of the velocity at this moment and multiply it by .
3. Adding all the products
we find an approximate value of the
length path L
traveled by a material point during 0,
T,
that is
.
( 6 )
4. Tending to zero we find the exact value of the length path L
.
( 7 )