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Electrical-A
Presented by……
Guidance by…..Vaishali G. mohadikarVinita G. Patel
Enrollnment No:
130940109040130940109044130940109050130940109043130940109044130940109045130940109046130940109052
Multiple integrals
Multiple Integrals
Double Integrals Triple Integrals
Cylindrical Coordinate
s
SphericalCoordinates
Double Integrals
Double integrals
Definition: The expression:
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.
2 2
1 1
( , ) .y x
y y x xf x y dx dy
m
1i
n
1jij
*ij
*ij0|P|
R
R
) ΔΔy,f(xlimy)dAf(x,
y)dAf(x, is R rectangle the over f of integral double The
Then, by Fubini’s Theorem,
( , ) ( , )
( , )
D R
b d
a c
f x y dA F x y dA
F x y dy dx
We assume that all the following integrals exist.
PROPERTIES OF DOUBLE INTEGRALS
, ,
, ,
D
D D
f x y g x y dA
f x y dA g x y dA
( ) ( ) ( )b c b
a a cf x dx f x dx f x dx
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:
1D
dA A DIf D = D1 D2, where D1 and D2 don’t overlap except perhaps on their boundaries, then
1 2
, , ,D D D
f x y dA f x y dA f x y dA
22
1-1
2
3
1-
1)x
2
1-xx
2
3x
4
1-x
2
1(
dx4x-x2
33x
2
32x-xx
)dx)(2x-)x((12
3)2x-xx(1
3y)dydx(x3y)dA(x
:Ans
}x1y2x 1,x-1|y){(x,D Where
3y)dA(x Evaluate 1.
:Example
5342
1
1-
44233
1
1-
222222
D
1
1-
x1
2x
22
D
2
2
62xy parabola theand 1-xy line the
by boundedregion theis D xydA where Evaluate 2.
2
D
36xydxdyxydA
4}y2- 1,yx2
6-y|y){(x,
}62xy? 5,x-3|y){(x,D
:Sol
D
4
2-
1y
2
6-y
2
2
b,ra|){(r,RConsider
Double Integrals in Polar Coordinates
Polar rectangle
D
)(h
)(h
21
R
b
a
2
1
)rdrdrsin ,f(rcosy)dAf(x,
thenDon continuous is f If region.
polor a be )}(hr)(h ,|){(r,DLet 2.
)rdrdrsin ,f(rcosy)dAf(x,
thenR,on continuous is f If 2-0 and rectangle
polar a be } b,ra|){(r,RLet 1.
Properties
2
15
)d7cos(15sin
)rdrd3rcos)(4(rsin3x)dA(4y
}0 2,r1|){(r,
4}yx1 0,y|y){(x,R
:Sol
4}yx1 0,y|y){(x,R e wher
3x)dA(4y Evaluate 1.
:Example
0
2
R0
2
1
22
22
22
R
2
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.
3
4
1243
7
32
33
7
33)2(
3
I
10
x-2yx:are limits the
line. strip horizontal a ake
2,1,2,0:
1,0,,0:
n.integratio oforder thechangingby y )()(: Evaluate 3.
1
0
443
1
0
3
32
1
0
33
3
2
21
0
3
2
1
0
222
2
1
1
0 0
2
1
2
0
2222
)2(
)2(2
)2(
xxx
xxx
xxx
xy
x
yx
RIRI
yxyx
dx
dxxdxy
)dydx(
x
T
yyyxx
yyyxx
dxd
x
x
-x
x
n
n
y y
Triple Integrals
Triple integrals
The expression:
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.
2 2 2
1 1 1
( , , ) . .z y x
z z y y x xf x y z dx dy dz
Determination of volumes by multiple integrals
The element of volume is:
Giving the volume V as:
That is:
. .V x y z
2 2 2
1 1 1
. .x x y y z z
x x y y z z
V x y z
2 2 2
1 1 1
. .x y z
x x y y z z
V dx dy dz
dydxz)dzy,f(x,z)dvy,f(x, then
y)}(x,φzy)(x,φ (x),gy(x)gb,xa|z)y,{(x,E If 2.
dAz)dzy,f(x,z)dvy,f(x, then
y)}(x,φzy)(x,φ D,y)(x,|z)y,{(x,E If 1.
properties
E
b
a
g
g
φ
φ
2121
E D
φ
φ
21
1(x)
1(x)
y)2(x,
y)1(x,
y)2(x,
y)1(x,
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 22 1 2 1
2 2
1 1 0 1 1
12 232 2
1 11
22
3 3
16
3
x y
x y z x y
x x
V dx dy dz dx x y dy
yx y dx x dx
:Sol
2z2y xand 0z 0, x2y, x
planes by the boundedon tetrahedr theof volume theFind 3.
3
1
2y)dydx-x-(22ydA-x-2V
}2
x-2y
2
x 1,x0|y){(x,D
D
1
0
2
x-2
2
x
2
)rdrdr-(1
)dAy-x-(1V
}20 1,r0|){(r,D:Sol
y-x-1z paraboloid theand
0z plane by the bounded solid theof volume theFind 2.
2
0
1
0
2
D
22
22
formula for triple integration in cylindrical coordinates.
E
h
h
rru
rrurdzdrdzrrfdVzyxf
)(
)(
)sin,cos(
)sin,cos(
2
1
2
1
),sin,cos(),,(
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=zx
y
D
22222.10 surfaces by the bounded solid theis D wheredV, Evaluate:Example ,z,zzyxyx
Here we use cylindrical co-ordinates(r,θ,z)∴ the limits are:
64
1
3
12
43
)1(
rdzdrdθrI
20
1r0
1zr i.e.
1
1
0
4320
2
0
1
0
2
2π
0
1
0
1
r
22
xr
r
yx
drdr
z
Formula for triple integration in spherical coordinates
E
dVzyxf ),,(
d
c
b
addpdppsomppf
sin)cos,sin,cossin( 2
where E is a spherical wedge given by
},,),,{( dcbpapE
0p 0
D
222222.1 sphere theof volumeover the dV Evaluate:Example x zyxzy
Here we use spherical co-ordinates (r,θ,z) ∴ The limits are:
5
4
5
122
5cos
sinI
20
0
10
1
0
5
020
2
0 0
1
0
22
r
rr ddrd
r
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