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BS01_10900_AE_C.pdf
1. DOUBLE INTEGRAL : REGIONS , LIMITS , REPEATED INTEGRAL
NAME :- SUBHAJIT NANDI
SECTION :- B
UNIVERSITY ROLL NO. :- 10900121089
STREAM :- COMPUTER SCIENCE AND ENGINEERING
SUBJECT CODE : - BSC-301
MATHEMATICS
2. CONTENTS
HISTORY RELATED TO DOUBLE INTEGRAL
INTRODUCTION TO VOLUME AND DOUBLE INTEGRAL
PROPERTIES OD DOUBLE INTEGRAL AND IMPORTANT FORMULAS
EVALUATION OF DOUBLE INTEGRAL
GEOMETRICAL AND PHYSICAL APPLICATIONS OF DOUBLE INTEGRAL
SOME EXAMPLES OF DOUBLE INTEGRALS
APPLICATION OF DOUBLE INTEGRAL IN ENGINEERING
ACKNOWLEDGEMNT
BIBLIOGRAPHY
3. HISTORY RELATED TO DOUBLE INTEGRAL
• Double integrals were calculated in the 18th and 19th centuries in the
same way they teach you in calculus now, using special cases of what later
became Fubini's theorem. Most of these special cases had no special
name.
• Actually double and triple integrals were calculated even before the notion of integral
was formalized, in antiquity, by people like Archimedes and Eudoxus, but they had to
invent a new argument for each particular integral. (They essentially approximated integrals
by finite sums and then tried to find the limit. The difficulty is in finding the limit
explicitly.)
• In the 17th century, Cavalieri's Principle was formulated which helps in evaluating
some multiple integrals, especially for areas and volumes. Cavalieri’s principle was still
mentioned in high school in the 1960s. But modern calculus books prefer to refer to a
very general Fubini theorem.
4. VOLUME AND DOUBLE INTEGRAL
We integrate a function f(x,y), called integrand, over a closed
bounded region R in the XY-plane, whose boundary curve has a
unique tangent at each point, but may have finitely many cusps (
such as vertices of a triangle or rectangle).
• We subdivide the region R by drawing parallel to the “x” and “y”
axes. We number the rectangles that are within R from 1 to n. In
each such rectangle, we choose a point, say in the
kth rectangle, and then we form the sum
5. • Where is the area of the kth rectangle. This we do for larger and larger
positive integers n in a completely independent manner but so that the length of
the maximum diagonal of the rectangles approaches zero as n approaches
infinity.
• In this fashion, we obtain a sequence of real numbers.
• Assuming that f(x,y) is continuous in R and R is bounded by
finitely many smooth curves, one can show that this sequence converges and its
limit is independent of the choice of subdivisions and corresponding points
This limit is called the DOUBLE INTEGRAL of f(x,y) over the
region R and is denoted by
6. PROPERTIES
• f(x,y) & g(x,y) continuous in a region R.
Furthermore, there exists at least one
point in R such that we have
Where A is the area of R; this is called the MEAN VALUE
THEOREM for double integrals.
7. EVALUATION OF DOUBLE INTEGRAL
(1)Suppose that R can be described by inequalities of the form
• represents the boundary of R . Then
R
d
c
a b
(2)Suppose that R can be described by
inequalities of the form
so that representsthe boundary of R . Then
8. GEOMETRICAL AND PHYSICAL MEANING
• The AREA A of a region R in the XY-plane is given by the double integral.
• The VOLUME V beneath the surface z= f(x,y)>0 and above a region
R in the XY-plane is
• Let f(x,y) be the density ( mass per unit volume) of a distribution of the
mass in the XY plane. Then the total mass M in R is
• The CENTER OF GRAVITY of the mass in R has the co-
ordinate where
&
• The MOMENT OF INERTIA of the mass in R about the “x” and
“y” axis respectively, are
10. APPLICATIONS OF DOUBLE INTEGRAL IN ENGINEERING
• Calculus is used in Computer Science for machine learning, data mining, scientific
computing, image processing, and creating the graphics and physics engines for video
games, including 3D visuals for simulations of real-life events. In addition to being used in a
wide range of software programs, Calculus is also used in a variety of other applications.
• Applications of multiple integrals arise whenever an area or a volume under a
2D curve (a surface) needs to be measured. Similarly, a triple integral can be
used to find the sum of some value in a solid (if you are given a density function
then the total mass of a solid can be calculated).
• By using Double integral we can find the 1) Charge of a Plate
2) Average Value of a Function
3) Computing Mass
11. ACKNOWLEDGEMENT
I would like to express my special thanks and gratitude to my teachers as well as our
supervisors and my University who gave me the golden opportunity to do this
wonderful project on the topic DOUBLE INTEGRAL, which also helped me in doing
a lot of research and I came to know about so many new things.
I am really thankful to them.
Secondly, I would also like to thank my friends who helped me in finishing this project
in a limited time.
It really helped me increase my knowledge and skills. Thanks To everyone who helped.
