The document discusses several real-world applications of differential and integral calculus. It provides examples of first-order differential equations being used to model jumping motions in video games and the cooling of objects. Surface and volume integrals are applied in fields like electrostatics, fluid dynamics, and continuity equations. Matrix determinants can estimate areas like that of the Bermuda Triangle. Overall, calculus has wide applications in science, engineering, economics and other domains.

1. KOTA
SESSION 2015-16
PRESENTATION
ON
SOME REAL WORLD APPLICATIONS OF DIFFERENTIAL &
INTEGRAL CALCULUS
SUBMITTED TO:- SUBMITTED BY:-
Mrs. Sona Raj Neha Nagar (K12236)
Ajay Singh (K12235)
Himanshi Choudhary (K12642)

2.  Introduction
 Example
 Application according to branch( computer
science)
 Real world application
 Conclusion
 Reference

3.  Applications of First order differential equation.
 Applications of Matrix and Determinants.
 Surface and Volume Integral.

4. FIRST ORDER DIFFERENTIAL EQUATION:
A first order differential equation involves only
the first derivative of the function. First-order
differential equation contain only dy/dx
equated to some function of x and y, and can
be written in either of two equivalent standard
forms. First-order means that only the first
derivative of y appears in the equation, and
higher derivatives are absent.

5.  A surface integral is a generalization of multiple integrals
to integration over surfaces. It can be thought of as the
double integral analog of the line integral.
 A volume integral refers to an integral over a three
dimensional domain, i.e., it is a special case of multiple
integrals. Volume integrals are especially important in
physics for many applications.
 the Divergence theorem, also known as Gauss’s theorem or
Ostrogradsky’s theorem, is a result that relates the flow
of a vector field through a surface to the behavior of the
vector field inside the surface.
 More precisely, it states that the outward flux of a vector
field through a closed surface is equal to the volume
integral of the divergence over the region inside the
surface, i.e., the sum of all sources minus the sum of all
sinks gives the net flow out of a region.

6.  Example 1: Solve
Answer :
Since the RHS of this equation can be factorized to
give x(1 + y), the equation becomes separable and
we obtain
Now integrating both sides, we get

7.  In the video games involving a jumping motion, a
differential equation is used to model the velocity of
a character after the command is given to return
them to the ground in a simulated gravitational field.
 It is widely used in algorithms which is a step by
step solution of a given problem.
 Differential equations is an essential tool for
describing the nature of the physical universe and
naturally also an essential part of models for
computer graphics and vision.

8.  Robotics- linear & angular motion of a mechanism, &
i.e. Also used to the define the relationship btw forces
applied on a mechanism & the torque needed to
support these forces.
 Machine learning- it includes computer vision. Also
involves solving for certain optimal conditions or
iterating towards a solution with techniques like
gradient descent or expectation maximization.
 Network analysis- any analysis regarding the
dynamics of a network is likely to involve the calculus.
To understand a outcome of a edge craetion model like
preferential attachment which says that nodes with
probability proportional to their existing degrees.

9.  Matrix also has its application in c programming
as a matrix multiplication problems in c
programming language
 Matrices are mainly used in 'Structural Analysis".
Suppose u need to analyze a multi storied frame (
to get the stresses & strains consequent to different
types of loads such as dead load, live load, wind
load ,seismic load etc.) we have to use " matrix
method' of frame analysis. Of different Matrix
method of analysis, direct stiffness method is used
in computers.

10.  Newton’s law of cooling
Newton's law of cooling states that if an object is hotter
than the ambient temperature, then the rate of
cooling of the object is proportional to the
temperature difference.
we introduced Newton’s law of cooling. The model
equation was dθ/dt = −k(θ − θs) where θ = θ(t) is the
temperature of the cooling object at time t, θs the
temperature of the environment (assumed constant)
and k is a thermal constant related to the object. Let
θo be the initial temperature of the liquid, i.e. θ = θo
at t = 0.

11.  Population Growth:
The population of a given species is decrease at a constant
rate of n people annum per emigration. The population
due to birth and death is increased at constant rate of
lambda % of the existing population per annum. If the
initial population is N people, then the population x
people after t time is given by
 Chemical Reaction:
If a temperature is kept constant, the velocity of chemical
reaction is proportional to the product of the
concentration of the substance which is reacting, then x
must satisfy the eq.
Where k is the positive constant, while a & b are two
reacting substances resp.

12.  Exponential Decay:
The rate of decay of a radioactive substance at a time t is
proportional to the mass x(t) of the substance left at that
time.
Thus,
is the positive constant.
 Spread of disease:
The equation considered is
where N is population that is infected at time t, P is the total
population which is susceptible to infection, and the rate
of change of N is assumed to be proportional to the
product of N & P-N.

13.  Physical laws-
Examples: Gauss’s law in electrostatics, magnetism &
gravity.
 Continuity equations-
In fluid dynamics, electromagnetism, quantum mechanics,
relativity theory & a number of other fields , there are
continuity equations that describe the conservation of
mass, momentum, energy, probability, or other quantities.
 Inverse square laws-
Two examples are Gauss' law, which follows from the
inverse-square Coulomb's law, and Gauss' law for gravity,
which follows from the inverse-square Newton's law of
universal gravitation. The derivation of the Gauss' law-type
equation from the inverse-square formulation (or vice
versa) is exactly the same in both cases.
 Integrals are also helpful in electrostatics, magnetism,
gravity & mass continuity.

14.  The Bermuda Triangle is a large triangular
region in the Atlantic ocean. Many ships and
airplanes have been lost in this region. The
triangle is formed by imaginary lines
connecting Bermuda, Puerto Rico, and Miami,
Florida. Use a determinant to estimate the area
of the Bermuda Triangle.
 Matrix is also used in cryptography.That to
encode this message “Meet Me At Fort”. After
encode we decode it which to read it in words.

15. The Bermuda Triangle is a large trianglular region in the Atlantic ocean.
Many ships and airplanes have been lost in this region. The triangle is
formed by imaginary lines connecting Bermuda, Puerto Rico, and Miami,
Florida. Use a determinant to estimate the area of the Bermuda Triangle.
E
W
N
S
Miami (0,0)
Bermuda (938,454)
Puerto Rico (900,-518)
.
.
.

16.  Essentially path integrals/function integrals
are used for quantum mechanics & for field
theories in physics.
 Not only in physics, calculus is used in
different fields of our day to day life & it
became an important part of professional jobs
& tasks.

17. www.quora.com
www.functionspace.com
www.teach-nology.com
www.m.reddil.com
www.academia.edu
• Books- H.K. Das
• N.P. Bali
• introduction to integral calculus
• introduction to integral calulus: systematic
studies with engineering applications