
- •Донецьк 2006
- •Integral calculus lecture no. 19. Primitive and indefinite integral
- •Point 1. Primitive
- •Properties of primitives
- •Point 2. Indefinite integral and its properties
- •Point 3. Integration by substitution (change of a variable)
- •Point 4. Integration by parts
- •Lecture no.20. Classes of integrable functions
- •Point 1. Rational functions (rational fractions)
- •Point 2. Trigonometric functions
- •Universal trigonometrical substitution
- •Other substitutions
- •Point 3. Irrational functions
- •Quadratic irrationalities. Trigonometric substitutions
- •Quadratic irrationalities (general case)
- •Indefinite integral: Basic Terminology
- •Lecture no. 21. Definite integral
- •Point 1. Problems leading to the concept ofa definite integral
- •Point 2. Definite integral
- •Point 3. Properties of a definite integral
- •I ntegration of inequalities
- •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 Change of a variable (substitution method)
- •Integration by parts
- •Lecture no.22. Applications of definite integral
- •Point 1. Problem – solving schemes. Areas
- •Additional remarks about the areas of plane figures
- •Point 2. Arс length
- •Point 3. Volumes
- •Volume of a body with known areas of its parallel cross-sections
- •Volume of a body of rotation
- •Point 4. Economic applications
- •Lecture no. 23. Definite integral: additional questions
- •Point 1. Approximate integration
- •Rectangular Formulas
- •Trapezium Formula
- •Simpson10 formula (parabolic formula)
- •Point 2. Improper integrals
- •Improper integrals of the first kind
- •Improper integrals of the second kind
- •Convergence tests
- •Point 3. Euler г- function
- •Definite integral: Basic Terminology
- •Lecture no. 24. Double integral
- •Point 1. Double integral
- •Point 2. Evaluation of a double integral in cartesian coordinates
- •Point 3. Improper double integrals. Poisson formula
- •Point 4. Double integral in polar coordinates
- •Double integral: Basic Terminology
- •Differential equations lecture no.25. First and second order differential equations
- •Point 1. General notions
- •Point 2. Integrable types of the first order differential equations (of de - 1)
- •1. Separated de-1 (de-1 with separated variables)
- •2. Separable de-1 (de-1 with separable variables)
- •3. Homogeneous de-1
- •4. Linear de-1
- •5. Bernoulli de-1
- •Point 3. Order reducing second order differential equations
- •Lecture no.26. Second order linear differential equations
- •Point 1. General notions
- •Point 2. Linear dependence and independence
- •Point 3. Homogeneous equations Structure of the general solution of so lhde
- •So lhde with constant coefficients
- •Point 4. Nonhomogeneous equations Structure of the general solution of so lnde
- •Method of variation of arbitrary constants
- •Method of undetermined coefficients for so lnde with constant coefficients
- •Lecture no. 27. Systems of differential equations. Approximate integration of differential equations
- •Point 1. Normal systems of differential equations
- •Point 2. Approximate integration of differential equations Successive approximations method
- •Euler method
- •Differential equations: Basic Terminology
- •Bibliography textbooks
- •Problem books
- •Contents
Point 2. Arс length
Let ˘ be an arc of some curve, and it’s necessary to find its length.
T
he
first method. We divide the arc ˜AB
into n
parts by points
and inscribe the poly-gonal line
in ˘AB (fig.
15). Let
Fig. 15
( 12 )
is
the perimeter of the polygonal line and
.
If there exists the limit
(
13 )
it is called the length of the arc ˘AB.
Let an arc ˘ of a curve is determined in Cartesian coordinates by an equation
( 14 )
on
a segment
,
and
be the coordinates of the point
,
.
In this case
,
and
by Lagrange theorem there is a point
such that
.
Denoting
we get
and therefore
.
Passage to the limit gives the desired arc length as a definite integral from a to b,
.
( 15 )
The arc length L exists if a function is continuous with the first derivative on the segment .
The second method. We find at first an element (or
the differential)
of
the desired arc length and then the arc length as the sum of all the
elements.
By Pythagorean theorem
and
.
( 16 )
For an arc ˘ determined by an equation (14)
,
and the sum of all the elements from a to b leads to the same formula (15).
If an arc ˘ of a curve is determined parametrically by equations
,
( 17 )
we have from (16)
,
and therefore
.
( 18 )
If an arc ˘ of a curve is given in polar coordinates by an equation
( 19 )
we pass to parametrical equations of the arc
( 20 )
and apply the formula (18). Since
the formula (18) gives
.
( 21 )
Ex. 10. Find the arc length of a curve
for
.
By virtue of the formula (15) we have
Ex. 11. Find the length of the loop of the curve
(fig. 11).
By the formula (18)
.
Ex. 12. Find the length of the cardioid (fig. 13).
With the help of the formula (21)
.
Ex. 13. Find the length of an ellipse
.
Parametrical equations of the ellipse
and
by the formula (18)
.
We can find only approximate value of L for given values of a and b because of a pri-mitive of the integrand is inexpressible in terms of elementary functions.
Ex. 14. Prove that the length of Bernoulli lemniscate (fig. 14) can be represented by the next integral
.