Point 5. Newton-leibniz formula
Theorem 3. If a function is continuous one on a segment a, b, and F(x) is one of its primitives, then Newton4-Leibniz5 formula for evaluation of the definite integral of the function over the segment a, b is true
( 23 )
■ We have two primitives: and the integral (21) with upper variable limit x. By corresponding property of primitives the difference
.
To find the value of the constant C we put . So
,
and therefore
.
Substituting x by b and t by x we obtain the formula (23).■
Note 1. The expression
,
which
means the action
,
is often called the double substitution.
Ex. 5. Calculate the definite integral
A primitive of is and therefore by Newton-Leibniz formula
Ex.
6. Find the area of a figure bounded by the next lines
,
(fig. 5).
The figure in question is a curvilinear trapezium, and so its area by the formula (10) equals the definite integral
.
Fig. 5 Ex. 7. A particle
moves in a straight line and t sec.
after passing a point O the
velocity of the particle is
ft. per sec. Find the distance of the particle from O
after 2 sec.
On the base of the formula (12) the distance in question equals
ft.
Ex. 8. Find the mean value of the function
on the segment
.
On the base of the formula (20)
.
Ex. 9. Find the mean value of the velocity
of the particle of the example 7 during the time interval from
to
.
By (20) (and taking into account the result of integration in ex. 7) we get
.
Point 6. Main methods of evaluation a definite integral Change of a variable (substitution method)
Theorem 4. Let: 1) a function is continuous on a segment [a, b]; 2) a function is continuous with its derivative on a segment [α, β]; 3) φ (α) = a, φ (β) = b. Then the next formula (formula of change of a variable) is true
.
( 24 )
■ Let
is some primitive of a function
.
Then
is the primitive of the function
.
By Newton-Leibniz formula
;
.■
Note 2. As distinct from an indefinite integral it isn’t necessary to return to the preceding variable after integration by the formula (24).
Ex. 9. Calculate the definite integral
.
Let’s put
.
Then
,
so
Ex. 10. Find the area of a figure bounded by an
ellipse
(fig. 5).
It’s sufficient to find the quadruplicated area of the Fig. 5 part OAB of the figure.
The first way. From the equation of the ellipse
,
and so
The second way. It’s better to pass to
parametric equations of the ellipse, na-mely
6.
In this case
.
Integration by parts
Theorem 5. If
functions
are continuous with their derivatives on a segment
,
then the next formula (formula of
integration by parts) is true
( 25 )
■To prove this formula it’s sufficient to integrate from a to b both parts of the identity
and
apply Newton-Leibniz formula for the integral of the expression
■
Ex. 11. Evaluate the definite integral
.
Ex. 12. Find the area of a figure boun-ded by two
curves
(see fig. 6).
The curves
inter-sect at the
points
and form a given figure
.
Its area is equal to the difference of the
areas of two curvilinear tra-
Fig. 6
peziums
.
.
Ex. 13. Let
.
Prove that
.
■
■
For example
.
Ex. 14. Find the remainder
of Taylor formula in Lagrange form.
Let, for example,
,
and
.
By Lagrange formula
.
Taking
we get
,
and
after integration over the segment
Therefore
To get for any n we write
,
then
we put
and integrate n times
over
.
As the result we’ll get
