SINGLE , DOUBLE , TRIPLE INTEGRAL
LEC :
2
11
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCES
-
:
PROPERTIES OF INTEGRAL
1
.
(
)
. ( )
.
( )
( )
0
( )
( )
( )
( )
( )
,
[ , ]
1
,
1
1
[ ( )
( )]
( )
( )
( )
0
( )
0
( )
b
b
a
a
b
b
a
a
a
a
b
a
a
b
b
c
b
a
a
c
n
n
b
b
b
a
a
a
b
a
dx
b
a
c dx
c b
a
c f x dx
c f x dx
f x dx
f x dx
f x dx
f x dx
f x dx
f x dx c
a b
x dx
x
C
n
n
f x
g x dx
f x dx
g x dx
if f x
f x dx
if f x
g
1
( )
( )
( )
[ ( )]
[ ( )]
( )
,
1
1
b
b
a
a
n
n
x
f x dx
g x dx
f x
f x
f
x dx
C
n
n
-
:
EXAM
10
6
10
6
3
3
2
1
1
10 6
4
5.
5(10 6)
5.4
20
1
.
(9 1)
4
2
2
dx
dx
x
x dx
c
b
a
f(x)
SINGLE , DOUBLE , TRIPLE INTEGRAL
LEC :
2
11
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCES
5
5
2
2
2
1
1
3
2
2
0
0
3
3
2
3
2
2
1
1
3
3
2
2
2
2
1
1
3
2
3
1
1
1
1
7
4
3
2
6
3
0
0
0
(
5)
5
25.5
2
3
11
(
3 )
3
2
6
(2
5 )
(2
5 ) (4
5)
?
3
1
(2
6 ) (2
3)
(2
6 ) (4
6)
2
(2
6 )
?
3
(
1)
(
2
1)
?
7
2
x
x
dx
x
x
x
x dx
x
x
x
x
x
x
dx
x
x
x
dx
x
x
x
dx
x
x
x
x
x
dx
x
x
dx
x
مــيـقـتسملا طـخلا ةــــلداعم
:
EQUATION OF STRAIGHT LINE
The general form of straight line equation is :-
0
ax
by
c
OR
y
mx
b
Find the equation of the curve whose slope at any point p(x,y) is 2x+1
-
:
EXAM
and passing through the point (1,3) .
2
2
2
2
1
(2
1)
(1,3)
3 1
1
1
1
dy
m
x
dx
dy
y
dx
x
dx
y
x
x
C
general curves
dx
curves
C
C
y
x
x
special curve
hose slope at any point p(x,y) is
Find the equation of the curve w
-
:
EXAM
3
2
4
18
8
3
x
x
x
and passing through the point (1,11) .
SINGLE , DOUBLE , TRIPLE INTEGRAL
LEC :
2
11
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCES
3
2
3
2
4
3
2
4
3
2
4
18
8
3
(4
18
8
3)
6
4
3
11 1 6
4 3
3
6
4
3
3
dy
m
x
x
x
dx
y
x
x
x
dx
y
x
x
x
x
C
C
C
y
x
x
x
x
DOUBLE INTEGRATION :
:
EXAM
1
0 0
1
1
1
2
2
2
2
0
0
0
0
1
2
3
0
(3
)
1
1
3
3
(3
)
(3
)
2
2
2
3
1
1
2
2
x
x
x
y dydx
y
xy
y
dx
x
x
x
dx
x
x
dx
x
x
:
EXAM
2
2
2
1
2
2
2
2
3
2
1
1
1
1
1
5
(
)
3
2
6
y
y
y
y
dxdy
x
dy
y
y dy
y
y
:
EXAM
2
2
2
2
3
2
4 2
2
0
4 2
2
2
2
4 2
2
2
2
4 2
0
0
0
2
2
0
4
2
4
2
2
4
2
2 (4
2
)
8
3
4
3
2
y
y
y
y
ydxdy
yx
dx
y
y
y
y dx
y
y dx
y
SINGLE , DOUBLE , TRIPLE INTEGRAL
LEC :
2
11
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCES
INTEGRATION :
TRIPLE
:
EXAM
2
2
2
2
1
0 0
1
1
2
2
0 0
0 0
1
1
3
2
2
2
3
2
3
3
0
0 0
0
1
1
1
4
3
4
3
4
3
5
4
0
0
0
(
1)
(
1)
(
1)
(
1)
1
1
3
3
1
1
4
4
4
1
9
3
3
3
3
15
3
15
x x
y
x
x
x
y
x
x
x
dzdydx
x
z
dydx
x
x
x
y dydx
x
x
xy
y dydx
x y
x y
xy
y
dx
x
x
x
x dx
x
x dx
x
x
:
EXAM
2
2
2
2
1
0 0
1
1
2
2
0 0
0 0
1
1
2
3
3
3
0
0
0
1
1
3
4
0
0
1
1
1
4
3
4
3
4
3
5
4
0
0
0
1
1
3
3
4
1
1
3
3
3
1
1
4
4
4
1
9
3
3
3
3
15
3
15
x x
y
x
x
x
y
x
dzdydx
z
dydx
x
y dydx
x y
y
dx
x
x dx
x dx
x
x
x
x
x dx
x
x dx
x
x
SINGLE , DOUBLE , TRIPLE INTEGRAL
LEC :
2
11
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCES
1) Find the equation of the curve whose slope at any point p(x,y) is
4
2
3
(
16
4) (
4)
m
x
x
x
and passing through the point (2,1) .
2) Find the equation of the curve whose slope at any point p(x,y) is
2
(
5)
m
x x
and passing through the point (2,1) .
3) Find :
2
1 3
2
0 0
1
0
1
1
0
1 2 3
3
2
0 0 0
(
2
4 )
(
)
y
x
y
x
x x
y
x
x
y dydx
y dxdy
x
y
z dzdydx
z y x dxdy dz
MOHAMED SABAH AL TAEE
M.SC / MATHEMATICS
E-MAIL : msmt_80@yahoo.com
2013 -2014