Electrical-A

Presented by……

Guidance by…..Vaishali G. mohadikarVinita G. Patel

Enrollnment No:

130940109040

130940109044

130940109050

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130940109052

Multiple Integrals

Double Integrals Triple Integrals

Cylindrical

Coordinates

Spherical

Coordinates

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

1j

i j

*

i j

*

i j0|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.

, ,

, ,

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 D

If 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

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

3

3

3

2

2

1

0

3

2

1

0

222

2

1

1

0 0

2

1

2

0

2222

)2(

)2(2

)2(

xxx

xxx

xx

xx

yx

yx

RIRI

yxyx

dx

dxxdxy

)dydx(

x

T

yyyxx

yyyxx

dxd

x

x

-x

x

n

n

y y

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=x

2+y

2tan θ= z=z

x

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

43

2

0

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

0

2

0

2

0 0

1

0

22

r

rr ddrd

r