5. Double integrals
Definition:
The expression:
y2
x2
y y1 x x1
f ( x, y )dx.dy
is called a double integral and provided the four limits
on the integral are all constant the order in which the
integrations are performed does not matter.
If the limits on one of the integrals involve the other
variable then the order in which the integrations are
performed is crucial.
6. T h e d o u b le t e g r ao f f o ve r t h e r e ct a n g le is
in
l
R
f (x ,y )d A
R
f (x ,y )d A
R
lim
|P|
0
m
n
i 1 j 1
f (xi*j, y i*j )Δ Δi j
7.
Then, by Fubini’s Theorem
,
f ( x, y ) dA
D
F ( x , y ) dA
R
b
d
a
c
F ( x, y ) dy dx
8.
We assume that all the following integrals exist.
b
a
f ( x) dx
f x, y
c
a
b
f ( x) dx
c
f ( x) dx
g x, y dA
D
f x, y dA
D
g x, y dA
D
9.
The next property of integrals says that,
if we integrate the constant function f(x, y) = 1 over a
region D, we get the area of D:
1 dA
A D
D
If D = D1 D2, where D1 and D2 don’t overlap except perhaps on their
boundaries, then
f x, y dA
D
f x, y dA
D1
f x, y dA
D2
10. Example :
1. Evaluate
(x
3y)dA
D
WhereD
Ans :
(x
{(x, y) | -1
3y)dA
1
x
1 x2
-1 2x 2
1, 2x 2
(x
y
1
x 2}
3y)dydx
D
3
x(1 x - 2x )
((1 x 2 ) 2 - (2x 2 ) 2 )dx
-1
2
1
3
3 4
x x 3 - 2x 3
3x 2
x - 4x 4 dx
-1
2
2
1 2 1 4 3
1 5 1
3
1
3
( x - x
x x - x )
1-1 2
2
4
2
2
2
1
2
2
2
11. 2. Evaluate
xydA w hereD is the region bounded by
D
x - 1 and the parabola y 2
theline y
2x
Sol :
D {(x, y) | -3 x 5, ? y
y2 - 6
{(x, y) |
2
xydA
D
4
y 1
-2
y 2 -6
2
x
2x 6}
y 1, - 2 y 4}
xydxdy 36
6
13. Properties
1. Let R {(r, ) | a
rectangle and 0
f(x, y)dA
r b,
} be a polar
2 If f is continuous on R, then
b
a
f(rcos , rsin )rdrd
R
2. Let D {(r, ) |
, h1 ( ) r
h 2 ( )} be a polor
region. If f is continuous on D then
f(x, y)dA
D
h2 ( )
h1 ( )
f(rcos , rsin )rdrd
14. Example :
(4y2
1. Evaluate
3x)dA
R
wher R
e
Sol :
R
{(x, y) | y
{(x, y) | y
0, 1 x 2
{(r, ) | 1 r
(4y
2
0, 1 x 2
y2
2, 0
3x)dA
0
15
2
4}
1
(4(rsin ) 2
R
(15sin 2
4}
}
2
0
y2
7cos )d
3rcos )rdrd
15.
Changing The Order of integration
Sometimes the iterated integrals with givan limits bocomes more
compliated.As we know that w.r.t. y, or may be integrated in the
reverse order.
If it is given first to integrate w.r.t. x,then to change it consider a
vertical strip line and determine the limits.
If it is given first to integrate w.r.t. y,then to change it consider a
horizontal strip line and determine the limits.
16. 1 y
(x
3. Evaluate :
22 y
2
2
y)
(x
0 0
I R :x
I R :x
1
n
1
0, x
y, y
n
2
0, x
2
0, y
y, y
2
2
y )dxdy by changing the order of integration.
0
1
1, y
2
Take a horizontal strip line.
the limits are : x y 2 - x
0
1 2 -x
I
(
2
x y
x 1
2
)dydx
0 x
y
1
x
2
y
3
2 x
x
dx
3
0
3
1
0
x
3
1
2x
2
0
2x
3
3
4
7x
3 4
7 3
3x
(2 x)
3
dx
4 1
(2 x)
4
3
12
0
2
(2
x)
(2 x)
3
3
x
x 3
3
dx
17.
18. Triple integrals
The expression:
z2
y2
x2
z z1 y y1 x x1
f ( x, y, z )dx.dy.dz
is called a triple integral and provided the six limits on
the integral are all constant the order in which the
integrations are performed does not matter.
If the limits on the integrals involve some of the
variables then the order in which the integrations are
performed is crucial.
19. Determination of volumes by multiple integrals
The element of volume is:
V
x. y. z
Giving the volume V as:
x x2 y y2 z z2
V
x. y. z
x x1 y y1 z z1
That is:
x2
y2
z2
V
dx.dy.dz
x x1 y y1 z z1
20. properties
1. If E {(x, y, z) | (x, y) D, φ1 (x, y)
then
φ 2(x,y)
f(x, y, z)dv
E
D
2. If E {(x, y, z) | a
then
f(x, y, z)dv
E
x
φ1(x,y)
z
φ 2 (x, y)}
f(x, y, z)dz dA
b, g 1 (x)
b
g1(x) φ 2(x,y)
a
g1(x) φ1(x,y)
y
g 2 (x), φ1 (x, y)
f(x, y, z)dzdydx
z
φ 2 (x, y)}
21. Example: Find the volume of the solid bounded by the
planes z = 0, x = 1, x = 2, y = −1, y = 1 and the surface z
= x2 + y2.
2
V
x2 y 2
1
dx
x 1
dy
y
2
x 1
16
3
dz
1
z 0
3
x2 y
2
y
3
1
x 1
1
x2
dx
y
2
2x2
dx
1
x 1
y 2 dy
1
2
dx
3
22. 3. Find the volume of the tetrahedron bounded by the planes
x 2y, x 0, z 0 and x 2y z 2
Sol :
D {(x, y) | 0 x 1,
x
2-x
y
}
2
2
V
2- x
2
x
0
2
2 - x - 2ydA
D
1
3
1
(2 - x - 2y)dydx
23. 2. Find the volume of the solid bounded by the plane z
and the paraboloid z 1 - x 2 - y 2
Sol : D {(r, ) | 0
r 1, 0
(1 - x 2 - y 2 )dA
V
D
2
1
0
0
2
(1 - r 2 )rdrd
2 }
0
24. formula for triple integration in cylindrical
coordinates.
f ( x, y, z )dV
E
h2 ( )
u 2 ( r cos , r sin )
h1 ( )
u1 ( r cos , r sin )
f (r cos , r sin , z )rdzdrd
To convert from cylindrical to rectangular
coordinates, we use the equations
1 x=r cosθ y=r sinθ z=z
whereas to convert from rectangular to
cylindrical coordinates, we use
2. r2=x2+y2 tan θ=
z=z
y
x
25.
26. 2
2
D
Here we use cylindrical coordinates(r,θ,z)
∴ the limits are:
x
y
i.e. r
0 r
0
2
z
1
z 1
1
2
2π 1 1
I
r
rdzdrdθ
0 0 r
2
1
r
2
(1
r ) drd
0 0
2
0
r
3
3
x
4
1
4
0
2
1
3
1
4
2
x y dV, where D is the solid bounded by the surfaces x y z
Example : Evaluate
2
2
6
2
,z
0,z 1.
27. Formula for triple integration in spherical coordinates
f ( x, y, z )dV
E
d
b
c
a
f ( p sin cos , p sin som , p cos ) p 2 sin dpd d
where E is a spherical wedge given by
E {( p, , ) a
p b,
,c
d}
29. x
Example : Evaluate
2
2
y z
2
dV over the volume of the sphere x
D
Here we use spherical co-ordinates (r,θ,z)
∴ The limits are:
0
0
r
1
0
2
2
1
2
r r
I
2
sin drd d
0 0 0
2
0
2
cos
2
1
5
0
r
5
1
5
4
5
0
2
2
y z
2
1.
