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1. LAXMI INSTITUTE OF TECHNOLOGY, SARIGAM
MULTIPLE
INTRIGRALS
calculus

2. Laxmi Institute of Technology ,
Sarigam
Branch: Mechanical Engineering
Div.: A
Subject : CALCULUS
Guided By : Mr. GAURAV GANDHI , Mr. DARPAN BULSARA
Prepared By : Roll no. 23 - 33

3. BRANCH - MECHANICAL-A
ACADAMIC YEAR – 2014-2015
SUBJECT - CALCULAS
SEM. – 1
 23- RAHUL SINGH
 24- SUDHANSHU DAS
 25- RAJAT SINGH
 26-ASHWIN GUPTA
 27- AMIT YADAV
 28- MUDDASIR KHAN
 29- VIVEK DUBEY
 30- ANKUR RANA
 31- FARUKH SHAIKH
 32- RAVINDRA YADAV
 33- AKSHAY AHIRE
 140860119097
 140860119019
 140860119
 140860119023
 140860119122
 140860119028
 140860119
 140860119
 140860119105
 140860119123
 140860119004

4. contents:
Double integral
Fubini’s theorem
Algebraic Properties
Double integrals in polar
coordinate
Triple integration

5. MULTIPLE INTEGRALS
DOUBLE INTEGRALS
 Let the function z=f(x,y) be defined on some region R
 Divide the region R into small parts by drawing lines parallel to X-axis and Y-axis within the region
R.
 These parts are in the form of rectangle of lengths x &
,except some of the parts at the boundaries of the region.
 Let the number of such rectangles be n. Each of area
 Choose any point in any area
 If we take then all the small parts wthin the region will approximately behave like
rectangles.
k k x y
 Then the following limit of sums over region R :

 y
( *, *)  A  x* y
n  
lim ( *, *)
1
n
n k k k
k
f x y  A

 

6. {( , , ) 0 ( , ),( , ) } 3 s  x y z R  z  f x y x y R

7. MULTIPLE INTEGRAL
DOUBLE INTEGRAL
 Is known as the double integral of f(x,y) over R and can be written as
 f (x, y)dA

8. MULTIPLE INTEGRAL
Fubini’s theorm
 If f(x,y) is continuous throught the rectangular region
 f ( x , y ) dA    f ( x , y )
dxdy
 f x y dA    f x y dydx
&
d b
R c a
b d
( , ) ( , )
R a c
d b d b
  f ( x , y ) dxdy   [  f ( x , y ) dx ]
dy
c a c a

9. MULTIPLE INTEGRAL
For example
2 1
  (13xy)dxdy
1 0
2 2
x y
1
0
I   x  dy
1
3
[ ]
2
2
I    dy
1
3
y
(1 )
2
2
2
1
3
y
 y 
[ ]
4
12 3
   
[2 1 ]
4 4
3
 4

4

10. MULTIPLE INTEGRAL
 Algebraic properties
Properties:
1. cf(x,y)dA 
c f(x,y)dA
 
R R
  
2. (f(x,y)  g(x,y))dA  f(x,y)dA 
g(x,y)dA
R R R
3. If f(x,y)  g(x,y)  (x,y) 
R, then

 

R R
R
f(x,y)dA g(x,y)dA
4 f(x,y) dA 0, if f (x, y) 0 on R
 

11. CHANGE TO POLAR COORDINATES
 Suppose that we want to evaluate a double
integral , where R is one of the regions
shown here

12. CHANGE TO POLAR COORDINATES
Double integrals in polar coordinate
 From this figure the polar coordinates (r, θ) of a point are
related
to the rectangular coordinates (x, y) by
the equations
r2 = x2 + y2
x = r cos θ
y = r sin θ
The regions in the first figure are special cases of a polar
rectangle
R = {(r, θ) | a ≤ r ≤ b, α ≤ θ ≤ β}
shown here.

13. CHANGE TO POLAR COORDINATES
 To compute the double integral
 where R is a polar rectangle, we divide:
 The interval [a, b] into m subintervals [ri–1, ri]
of equal width Δr = (b – a)/m.
The interval [α ,β] into n subintervals [θj–1, θi]
of equal width Δθ = (β – α)/n.

14. CHANGE TO POLAR COORDINATES
Then, the circles r = ri and the rays θ = θi
divide the polar rectangle R into the small
polar rectangles shown here

15. CHANGE TO POLAR COORDINATES
 The “center” of the polar subrectangle
Rij = {(r, θ) | ri–1 ≤ r ≤ ri, θj–1 ≤ θ ≤ θi}
has polar coordinates
ri* = ½ (ri–1 + ri)
θj* = ½(θj–1 + θj)

16. CHANGE TO POLAR COORDINATES
The rectangular coordinates of the center
of Rij are (ri* cos θj*, ri* sin θj*).

17. CHANGE TO POLAR COORDINATES

18. Triple integrals
 We consider a continuous function f(x,y,z) defined over a
bounded by a solid region D in three dimensional space .We
divide a solid region D into small rectangular parallelepiped
by drawing planes parallel to coordinate planes .
 Consider any point in one of the cells with
volume
 If we partition the solid region D into large number of such cells
then the following limits of the approximation of sum
Vk  xk . yk . zk 

19. Triple integrals
 Is known as triple integral of f(x,y,z) over D and can be written as:
Example 1
Evaluate the triple integral
where B is the rectangular box

20. Triple integrals

21. DEFINITION The Jacobian of the transformation T
given by x= g (u, v) and
y= h (u, v) is
y

u
x

v
y

v
x

u
x

y
v
x

y

u
v
u
x y

( , )
( , )
u v














22. CHANGE OF VARIABLES IN A DOUBLE INTEGRAL Suppose
that T is a C1
transformation whose Jacobian is nonzero and that maps
a region S in the uv-plane onto a region R in the xy-plane.
Suppose that f is continuous on R and that R and S are type
I or type II plane regions. Suppose also that T is one-to-one,
except perhaps on the boundary of . S. Then


( , )
  
R S
dudv
x y
u v
f x y dA f x u v y u v
( , )
( , ) ( ( , ), ( , ))

23. Let T be a transformation that maps a region S in
uvw-space onto a region R in xyz-space by means of
the equations
x=g (u, v, w) y=h (u, v, w) z=k (u, v, w)
The Jacobian of T is the following 3X3 determinant:

24. 
R
x y z

( , , )
( u , v , w
)

f (x, y, z)dV
x

w
y


w
z


w

x

v

y

v

z

v
x

u
y


u
z


u




 

s
dudvdw
x y z
( , , )
u v w
f x u v w y u v w z u v w
( , , )
( ( , , ), ( , , ), ( , , ))