DR.S.&S.S . GANDHY GOVERNMENT ENGINEERING SURATS U B = C A L C U LU ST O P I C = M U LT I P L E I N T E G R A LP R E PA R E D BY: -1. C - 2 7 P R A S H A N T A C H E L A N I 2. C - 4 0 V I J AY D VA D H E R3. C - 4 3 N I K H I L R PAT I L4. C - 2 1 P R AT I K B PAT E L

IN THIS :

1 Double Integrals over Rectangles 2 Double Integrals over General Regions 3 Double Integrals in Polar Coordinates 4 Applications of Double Integrals 5 Triple Integrals 6 Triple Integrals in Cylindrical Coordinates 7 Triple Integrals in Spherical Coordinates 8 Change of Variables in Multiple Integrals Review

function. continuous twoare h,h whered},yc (y),hx(y)h|y){(x,D

if II typeof be tosaid is Dregion planeA 3.function. continuous twoare g ,g where

(x)},gy(x)g b,xa|y){(x,D if I typeof be tosaid is Dregion planeA 2.

y)dAF(x,y)dAf(x,

is Dover f of integral double The 1.:Definition

21

21

21

21

D R

d}yc (y),hx(y)h |y){(x,D where

y)dxdyf(x,y)dAf(x,

thenDregion II typeaon continuous is f If 2.

y)dydxf(x,y)dAf(x, then

(x)},gy(x)g b,xa|y){(x,D such that Dregion I typeaon continuous is f If 1.

:PropertiesII Type },y1x2y 1,y-1|y){(x,D 2.

I Type 1},ysinx ,x0|y){(x,D 1.:Example

21

D

d

c

(y)h

h

D

b

a

(x)g

g

21

222

1

2

1(y)

2

1(x)

221-1

23

1-1

)x21-xx

23x

41-x

21(

dx4x-x233x

232x-xx

)dx)(2x-)x((123)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

DOUBLE INTEGRALS IN POLAR COORDINATES

b,ra|){(r,RConsider

Polar rectangle

3. If f is continuous on a polar region of the form

then

)()(,),{( 21 hrhrD

D

h

hrdrdrrfdAyxf

)(

)(

2

1

)sin,cos(),(

215

)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 ere wh

3x)dA(4y Evaluate 1.

:Example

0

2

R0

2

1

22

22

22

R

2

TRIPLE INTEGRALS:The triple integral is defined in a similar manner to that

of the double integral if f(x,y,z) is continuous and single-valued function of x, y, z over the region R of space enclosed by the surface S. We sub divide the region R into rectangular cells by planes parallel to the three co-ordinate planes(fig 1).The parallelopiped cells may have the dimensions of δx, δy and δz.We number the cells inside R as δV1, δV2,…..δVn.

.

In each such parallelopiped cell we choose an arbitrary point in the k th

pareallelopiped cell whose volume is δVk and then we form the sum

=

.

.

TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES:

We obtain cylindrical coordinates for space by combining polar coordinates (r, θ) in the xy-plane with the usual z-axis.

This assigns every point in space one or more coordinates triples of the form (r, θ, z) as shown in figure.2.

.

DEFINITION : CYLINDRICAL COORDINATE Cylindrical coordinate represent a point P in space by orders triples (r, θ, z) in

which 1. (r, θ) are polar coordinates for the vertical projection of P on xy-plane. 2. z is the rectangular vertical coordinates.

The rectangular (x , y , z) and cylindrical coordinates are related by the usual equations as follow :

x = r cosθ, y = r sinθ , z = z = + , tanθ =

FORMULA FOR TRIPPLE INTEGRAL IN CYLINDRICAL COORDINATES

where,volume element in cylindrical coordinates is given by dV = rdzdrd

TRIPLE INTEGRALS IN SPHERICAL CO-ORDINATES: Spherical coordinates locate points in space is with two angles and one distance, as shown in figure.3. The first coordinate P = |OP|, is the point’sdistance from the origin.The second coordinate ф, is the angle OP make with the positive z-axis.It is required to lie in the interval 0 ≤ ф ≤ π.

The third coordinate is the angle θ as measured in cylindrical coordinates.

FIGURE.3

DEFINITION : SPHERICAL COORDINATES Spherical coordinates represent a point P in ordered triples (ƍ , θ , ф) in which 1. ƍ is the distance from P to the origin. 2. θ is the angle from cylindrical coordinates. 3. ф is the angle OP makes with the positive z-axis (0 ≤ ф ≤ π). The rectangular coordinates (x , y, z) and spherical coordinates are related by the following equations : x = ƍ sinф cosθ , y = ƍ sinф sinθ, z = P cosф.

FORMULA FOR TRIPLE INTEGRAL IN SPHERICAL COORDINATES:-

where, D = {(ƍ , θ , ф) | a ≤ ƍ ≤ b, α ≤ θ ≤ β, c ≤ ф ≤ d} and dV = dƍdф.

THANK YOU