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51
Figure 5
Fig.5 – graphic representation of area between two lines.
The search area (shaded in the figure 4) can be found by the formula:
2
1
2
2
òò
1
é
xdxdxxS
ê
ë
23
2
ù
xx
-=-=
23
8
ú
3
1
û
1
4
3
2
5
1
( square units)
=+--=
6
2
Volumes by slices
b
The volume of the solid can be found by the formula:
=
ò
a
.
dxxQV )(
Figure 6.
Fig.6 – graphic representation of a volume of the solid.
52
Figure 6 shows one basic way – using slices. The slices have thick-
=
3
2
yx
3
2
yx
1
2
2
x
dx
1(1)(101)9
+=×+=-»
0
coslimcoslimsin
b
xdxxdxx
ness dx and the area Q of a cross-section. This area is different for different slices: Q depends on x.
The volume of solid of revolution.
The volume of that solid is made easier because every cross-section is
a circle. We add the slices:
b
volume of solid of revolution
2
p=
ò
a
dxxfV )(
Length of a Plane Curve.
The graph of
is a curve in the x-y plane. How long is that
=
curve?
The length of curve:
b
( )
+=
ò
a
2
¢
dxxfS
)(1
A definite integral needs endpoints, and we specify x = 0 and x = 4.
Keep
4
ò
0
=
92498
xdxx
439427
and
dy
=
.
4
333
222
0
3
Improper integrals.
b
Definition: If there is a limit
ò
¥®
b
a
, then this limit is called the
dxxf )(lim
improper integral of the function f(x) on the interval [a, ¥).
If this limit exists and is equal to the constant, then we say that the
improper integral converges.
Example.
¥
òò
00
lim(sinsin0)limsin
=-=
bb
®¥®¥
b
==
bb
®¥®¥
bb
(does not exist).
Improper integral diverges.
53
Example.
1
-
dx
2
x
¥-
1
-
dx
limlim
2
òò
-¥®
b
x
b
-¥®
b
1
-
1
ù
é
-==
ú
ê
û
ë
b
æ
1lim
+=
ç
-¥®
b
è
verges.
Reading questions
1. What is the area between the two curves?
2. When the improper integral converges?
3. Where apply the definite integral?
1
ö
– Improper integral con-
1
=
÷
bx
ø
54
Section 5. Integral calculation
fxydydx
òò
(,)
fxydy
ò
of multi-variable functions
Topic 17. Multiple integral
Introduction. This topic shows how to integrate functions of two or
more variables. The key idea is to replace a double integral by two ordinary "single" integrals.
The double integral
(,)
fixed x we integrate with respect to y. The answer depends on x. Now
integrate again, this time with respect to x. The limits of integration need
care and attention! Frequently those limits on y and x are the hardest
part.
If f(x, y) is continuous on the closed area D is bounded by lines х = a,
x = b, (a < b), y = j(x), y = y(x), где j и y – there are a continuous
function, then
y
)(
b
æ
ç
=
ç
jD
)(
axx
è
Example. To calculate the integral
limited to the lines: y = 0, y = x2, x = 2.
starts with
ö
÷
=
÷
ø
- dxdyyx )(
òò
D
y
)(
x
b
òòòòòò
j
)(
x
a
, if the scope D is
. For each
),(),(),(
dyyxfdxdxdyyxfdxdyyxf
Figure 7
Fig.7 – graphic representation of the scope D what is limited to the
lines: y = 0, y = x2, x = 2.
55
2
2
0
(,)()()
yx
=
=
=
-
xyzdzdydxxydydxxyxydydxxydydx
====
xydxdy
òò
22
x
fxydxdydxxydyxy
ттттт
D
2
3
()
xdx
=-=-
ò
0
=
=-=-
000
445
xxx
2410
æö
ç÷
èø
8,02,34
2
0
2
y
=
2
y
Triple Integrals.
The only difference is that when the triple integral integration is not
in two and three variables, and the area of integration is not part of the
plane, and some area in three dimensional space:
xxyyz
21212
=
òòòòòò
zr
1
Example. To calculate the integral
2222
xy
1111
xxxx
2222243
ттттттттт
000000000
111
111111
xdxdxxdxx
òòò
24248813104
000
2
448
x
æö
yxx
41213
ç÷
èø
===×=
00
2
xy
æö
z
ç÷
222
0
èø
),,(),,(
dzdydxzyxfdxdydzzyxf
1
x
òòò
000
11
.
2
xy
2
yzdzdydxx
1
.
Reading questions
1. How to integrate functions of two variables?
2. How to integrate functions of three variables?
3. To calculate the integral
()
+
D
, if the scope D is limited
to the lines: y = 0, y = x2, x = 2.
Topic 18. Contour and surface integrals
Introduction. This lecture is devoted to the discussion of the following concepts: a line contour integral, calculation of work along a curve
and flow across a curve, the surface integral.
56
A line contour integral is an integral along a curve. It can equal an ar-
FTds
×
Fnds
×
ò
FTds
×
ò
xy
Fij
=+
FTds
×
()()
fQfP
-
FTds
×
dxidyj
×+×
FTdsijdxidyjdxdydf
¶¶¶¶
×=+×-=+=
()()
dffQfP
ò
ea, but that is a special case and not typical. Instead of area, here are two
important line integrals in physics and engineering:
Work along a curve
Flow across a curve
ò
c
c
In the first integral, F is a force field. In the second integral, F is a
flow field.
Work is done in the direction of movement, so we integrate F T. Here
T is the unit tangent vector, and F T is the force component along the
curve. Similarly n is the unit normal vector, at right angles with T. Then
F n is the component of flow perpendicular to the curve.
Most line integrals depend on the path. Those that don't are crucially
important. For a gradient field, we only need to know the starting point P
and the finish Q.
Theorem. When F is the gradient of a potential function f (x, y), the
work
c
depends only on the end points P and Q. The work is the
change in f
ff
If
May I give a rough explanation of the work integral
¶¶
¶¶
then
ò
c
=
ò
c
? It be-
comes clearer when the small movement Tds is written as
The work is the dot product with F:
ffff
()()
xyxy
¶¶¶¶
The infinitesimal work is df: The total work is
=-
This is the Fundamental Theorem for a line integral. Only one warning: When F is not the gradient of any f, the Theorem does not apply.
A line contour integral along the length of the arc AB will be according to the formula:
AB
b
= dttztytxtztytxfdszyxf
òò
a
¢
¢
+
222
¢
+
)()()())(),(),((),,(
.
57
Example. To calculate the integral
p££==
=
££j
=
()()
dffQfP
ò
screw
2
AB
æ
4
p
ç
122
+p=
ç
3
è
0
2
ö
÷
.
÷
ø
ò
AB
ttztytx
222
one turn of the
++
dszyx )(
.20;;sin;cos
2
22222222
pp
òò ò
0
If the integration is performed along the length of the plane curve
given by the equation
b
òò
aAB
then we get:
,),( bxaxy
2
j¢+j=
dxxxxfdsyxf )(1))(,(),(
The surface integral is a generalization of the double integral, which
line contour integral is with respect to a definite integral.
2
)1(21cos)sin()sin(cos)(
=+=++-++=++
dttdttttttdszyx
Fig.8 – graphic representation of the integral over the surface area.
Consider a surface in space, which is randomly divided into n parts
and the integral over the surface area is
Reading questions
1. What integral does respect to a surface integral?
2. Do most line integrals depend on the path?
3. What does it mean "Q" in the formula
òò
Figure 8
n
å
®l
0
=
i
S
1
=-
.
Dgba=
SFdSzyxF
),,(lim),,(
iiii
?
58
Section 6. Differential equations
=
¢
dx
dx
=¢¢
¢
=+¢
dx
-
=
=
+
x
Topic 19. Basic differential equations
Introduction. The solution of geometrical, physical and engineering
problems often lead to equations that relate the independent variables to
any function of these variables and the derivatives of this function.
Definition. If the differential equation has one independent variable,
it is called an ordinary differential equation
Definition. Higher order derivatives included in the equation, is
called the order of the differential equation.
Example.
3
¢
xyyx
order. In General form is written
2
x =++
dy
yd
xy
2
der. In General form is written
Definition. The integral of a differential equation is any equation that
does not contain derivatives, for which the differential equation is a consequence.
Example. Find the General solution of the differential equation
.
0
yyx
The General solution of the differential equation is by integrating the
left and right parts of the equations, which are already converted as look
like this:
dy
x
; ydxxdy
0=+ y
Now integrate:
lnln Cxy +-=
C
y = is the General solution of the original differential equation.
0
This is an ordinary differential equation of 1st
058
=+-+
.
0),,(
yyxF
2
– ordinary differential equation of the 2nd or-
yx
0),,,(
yyyxF
y
;
ln Cxy =
0
dx
.
-=
x
C
0
; Cexy
0
==
dy
ò ò
y
;
dy
;
dx
;
-=
x
lnln Cxy
;
59
Example. Find the General solution of the differential equation:
=+¢
dx
+-=
=
+
ò
=
+
ò
2
.0
yy
Cx
dy
-=
x
-
.
eCy
×=
1
dx
-=
y
dy
;
y
;
dy
-= dx
ò ò
y
;
ln
;
Cxy
-
eey ×=
Next, let us consider the techniques and methods used in the solution
of the differential equations of various types.
0),(),(
dyyxQdxyxP
- this is the so-called differential form of the equation of the first or-
der.
Equation of the form y’ = f(x).
Let the function f(x) is defined and continuous on some interval a < x
< b. In this case, all solutions of this differential equation are as
Cdxxfy +=
)(
.
Equations with variables which can be divided.
Separable differential equation — In mathematics, a separable differential equation may refer to one of two related things, both of which are
differential equations that can be solved by a method of separation of
variables. (For ordinary differential equations).
Definition. A differential equation y' = f(x, y) is called an equation
with separable variables if it can be written in the form y' = α(x)β(x).
This equation can be represented in the form:
;0)()(
dyyYdxxX
;
CdyyYdxxX =+
)()(
òò
.
After finding the relevant integrals obtained the General solution of
the differential equation with multiple variables.
If you have set of initial conditions, then they can be substituted into
the General solution is a constant value and, respectively, you get a particular solution.
Example. To find the solution of the differential equation
y
y
ydx
dy
¢
y
ln=
given the initial conditions у(2) = 1.
ydy
ln=
;
dx
y
;
ln
dx
=
y
ydy
ln
=
òò
y
=+ )(lnln yydCx
;
;
ln2y
Cx =+
If у(2) = 1
;
60
2
2
=
+
get ;2;02;
2
1ln
-=Þ=+Þ=+ CCC
42 -±=x
ey
– is a particular solution;
So:
2
;ln)2(2
yx =-
or
Homogeneous equations.
Definition. The function f(x, y) is called homogeneous of the n – th
dimension relative to its arguments x and y, if for any value of t (except
n
zero) is the the following equation:
=
).,(),( yxfttytxf
Example. To prove that the function is homogeneous
23
?3),(
yxxyxf +=
3233233323
yxftyxxtyxtxttytxtxtytxf =+=+=+=
Thus, the function f(x, y) is a homogeneous of the third order.
0),(),(
Any equation of the form
dyyxQdxyxP
is homogeneous if
the function P(x, y) и Q(x, y) are homogeneous functions of the same
dimension.
The solution to any homogeneous equation is based on bringing this
equation to an equation with separable variables.
Reading questions
1. Can the homogeneous differential equation contain a homogeneous
functions of different orders?
2. Do differential equations contain independent variables to any
function of these variables and the derivatives of this function?
3. What types of differential equations have you learned from this
lecture?
Topic 20. Liner differential equations
Introduction. In mathematics uses many types of linear differential
equations. in this lecture we will focus on the study of linear homogeneous differential equations with constant coefficients.
Linear homogeneous differential equation with constant coefficients
is an equation of the form 0...
)(
)1(
-
nn
1
=+++
yayay
n
.
),()3(3)(3)(),(
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