multiple intigration ppt
TRANSCRIPT
![Page 1: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/1.jpg)
Electrical-A
Presented by……
Guidance by…..Vaishali G. mohadikarVinita G. Patel
Enrollnment No:
130940109040130940109044130940109050130940109043130940109044130940109045130940109046130940109052
![Page 2: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/2.jpg)
Multiple integrals
![Page 3: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/3.jpg)
Multiple Integrals
Double Integrals Triple Integrals
Cylindrical Coordinate
s
SphericalCoordinates
![Page 4: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/4.jpg)
Double Integrals
![Page 5: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/5.jpg)
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
![Page 6: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/6.jpg)
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
![Page 7: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/7.jpg)
Then, by Fubini’s Theorem,
( , ) ( , )
( , )
D R
b d
a c
f x y dA F x y dA
F x y dy dx
![Page 8: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/8.jpg)
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
![Page 9: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/9.jpg)
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
![Page 10: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/10.jpg)
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
![Page 11: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/11.jpg)
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
![Page 12: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/12.jpg)
b,ra|){(r,RConsider
Double Integrals in Polar Coordinates
Polar rectangle
![Page 13: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/13.jpg)
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
![Page 14: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/14.jpg)
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
![Page 15: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/15.jpg)
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.
![Page 16: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/16.jpg)
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
![Page 17: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/17.jpg)
Triple Integrals
![Page 18: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/18.jpg)
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
![Page 19: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/19.jpg)
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
![Page 20: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/20.jpg)
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,
![Page 21: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/21.jpg)
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
![Page 22: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/22.jpg)
: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
![Page 23: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/23.jpg)
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
![Page 24: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/24.jpg)
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
![Page 25: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/25.jpg)
![Page 26: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/26.jpg)
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
![Page 27: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/27.jpg)
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
![Page 28: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/28.jpg)
0p 0
![Page 29: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/29.jpg)
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
![Page 30: Multiple intigration ppt](https://reader035.vdocuments.mx/reader035/viewer/2022062708/5589602cd8b42a52718b46a9/html5/thumbnails/30.jpg)
THANK YOU