This document contains information about a calculus project completed by students of the Mechanical Engineering department at Laxmi Institute of Technology in Sarigam. It includes the names and student IDs of 13 students who participated in the project. The document covers topics in multiple integrals, including double integrals, Fubini's theorem, double integrals in polar coordinates, and triple integrals. Formulas and examples are provided for each topic.

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
( , , )
( ( , , ), ( , , ), ( , , ))