3. Introduction:-
In elementary calculus, students have studied the
definite integral of a function of one variable . In this
chapeter we show how the notion of the definite integral
can be extended to functions of several variables. In
particular, we shall discuss the double integral of a
functions of two variable and the triple integral of a
function of three variables.
4. Define double integrals:-
Let f(x,y) be a function of two independent
variables x,y in - plane defined at all points in a finite
region A. let the region A be divided into n subregions
(k=1,2,…,n) of areas
let be any point inside the kth element
Let us form the sum
let us the number of subdivision becomes
infinite in such a way that the dimensions of each subdivision
approaches to zero.
kR nAAA ,....,, 21
kk , kR
n
k
kkk Af
1
),(
xy
5. If under these conditions, the limit
which is independent of the way in which the point
are chosen exists, then this limit is called the double
integral of over the region A, written
is defined by
n
k
kkk
n
Af
1
),(lim
kk ,
),( yxf
A
dxdyyxf ),(
kk
n
k
k
A n
Afdxdyyxf
),(lim),(
1
6. Properties of double integral:-
if f and g are continuous over the bounded
region R, then:
where R is composed of two subregions R1 and
R2 or
R R
dAyxfkdAyxkfP ),(),(.
1
R RR
dAyxgdAyxfdAyxgyxfP ),(),(),(),(.2
R R R
dAyxfdAyxfdAyxfP 1 23
),(),(),(.
21 1 2
),(),(),(
RR R R
dAyxfdAyxfdAyxf
dxdydA
8. Example 2- when the region of integration R is the triangle
bounded by y= 0, y= x and x = 1 , show that
sol. The region of interation is shown shaded in the
adjoining figure. Let us divide the triangle OAB into
vertical strips. Then it is evident that in an elementary
strips y varies from y = 0 to y = x while x varies from
x = 0 to x = 1
).
2
3
3
(
3
1
²)²4(
R
dxdyyx
10. Thus the given doudle integral can be expressed as the
repeated integral
dxdyyx
R ²)²4(
dxdyyx
x
x
y
1
0 0
²)²4(
x
y
x x
y
xyxy
0
1
0
1
2
sin².4.
2
1
²)²4(
2
1
dx
x
x
xxxx
x
2
sin²4²)²4(
2
1 1
1
0
a
x
axax 1
sin²..
2
1
²)²(
2
1 dxxa ²)²(
1
0 0
²)²4(
x
x
y
dxdyyx yxxa ,2
12. DEFINE TRIPLE INTEGRALS0:-
Let be a function
of three independent variables x,y,z defined for all points
in a finite closed three dimensional region V of space.
Divide V into n sub- regions of volumes , k= 1,2,….,n.
let us select an arbitrary point in each
and form the sum
let the number of sub-division become infinite in such a
way that the maximum dimensions of each
approaches to zero.
zyxf ,,
kV)( ,, kkkkP
kV
k
n
k
kkk V1
,, )(
kV
13. If under these conditions, the limit
Exists, which is independent of the way in which the points
are chosen, then this limit is called the triple
integral of over the region V, written
is defined by
kkk
n
k
k
n
Vf
)( ,
1
,lim
)( ,, kkk
),,( zyxf
,),,( dVzyxf
v
dVzyxf
v ),,(
n
k
kkkk
n
Vf
1
, )(lim
14. Example of triple integrals:-
(1).Evaluate:
Sol:- let the given triple integral be denoted by . Then
3
0
2
0
1
0
)( dxdydzzyx
I
3
0
2
0
1
0
)( dxdydzzyxI
dxdydzzyx
3
0
2
0
1
0
)(
dxdy
z
yzxz
3
0
2
0
1
02
²
dxdyyx
3
0
2
0
0
2
²1
16. (2).evaluate:-
where the region of integration V is a cylinder,which is
bounded by the following surfaces:
z = 0, z = 1,x²+y² = 4
Sol:-
form the adjoining figure it is evident that in the region
of integration V, z varies from z = 0 to z = 1, y varies from
y = to y = and x varies from x = -2 to
x = 2
dxdydzz
v
²)4( x ²)4( x