This document certifies that Mr. Shiv Kumar from St. John's Academy has successfully completed a math project on the applications of integrals, demonstrating sincerity and adherence to guidelines. It covers the fundamentals of integrals, including types, properties, and practical applications in various fields such as geometry and physics. The conclusion emphasizes the importance of integrals in understanding change and accumulation, revealing their relevance in real-world problem-solving.

1. ST. JOHN‘S ACADEMY
Submitted by:
Name:
Class: XII
Roll No:
Date:

2. Applications
of
Integrals

3. CERTIFICATE
This is to certify that Mr. Shiv Kumar of
Class 12th of St. John’s Academy Roll
No. __ has successfully completed his Maths
project file. He has taken proper care and
utmost sincerity in completion of his project.
All the work related to the project was done by
the candidate himself. The approach towards
the subject has been sincere and scientific.
I certify that this project is upto my
expectations and as per the guidelines
issued by the CBSE.
________________
________________ ________________
Student
Maths Teacher Principal

4. Acknowledgement
This acknowledgment is dedicated to all the
individuals who have provided valuable resources
and assistance in creating this project file. I
would like to thank my school authorities for
their constant support and the library staff for
granting access to relevant research material.
Their contributions have played a significant role
in making this project comprehensive and
informative.

5. 1-5 Integrals
Symbols
Types of Integrals
Basic formulae
Properties of Integrals
Fundamental Theorems
6-10 Applications of Integrals
Introduction
Area under simple curves
11 Conclusion

6. Integrals
In mathematics, integrals refer to a fundamental
concept in calculus that represents the accumulation
or total of a quantity. It involves finding the
antiderivative (reverse operation of differentiation) of
a function, and it is often used to calculate areas under
curves and solve problems related to accumulation or
change.
Symbol:
The symbol of integration is ∫, and it represents the
process of finding the integral of a function. It is used in
calculus to denote the antiderivative, or the reverse
process of differentiation.

7. Types of Integrals
There are two types of Integrals:
2.Definite Integral:
Symbol: ∫[a, b] f(x) dx
Meaning: Represents the accumulated quantity of a
function f(x) over a specified interval [a, b].
Example: ∫[0, 1] x^2 dx represents the area under the curve
of the function x^2 from x = 0 to x = 1.
1.Indefinite Integral:
Symbol: ∫ f(x) dx
Meaning: Represents the antiderivative of a function,
or the general solution to a differential equation.
Example: ∫ 2x dx = x^2 + C, where C is the constant of
integration.

8. BASIC FORMULAE

9. Properties of Indefinite Integrals.
∫ x^n dx = x^(n+1) / (n+1) + C, where n ≠ -1
4. Power Rule:
1. Sum and Difference Rule:
3. Integration of a Constant:
2. Constant Multiple Rule:
∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
∫ k f(x) dx = k ∫ f(x) dx, where k is any constant
∫ k dx = kx + C, where k is any constant
Properties of Definite Integrals.
1. Additivity:
2.Constant Multiple Rule
3. Interval Splitting
4. Change of Limits
5. Symmetry:

10. Fundamental Theorem
of Integral Calculas
First Fundamental Theorem of Calculus:
Second Fundamental Theorem of Calculus:

11. Application of
integrals
There are many applications of integrals out of
which some are mentioned below.
In maths:
. To find the center of mass (centroid) of an area
having Curved sides
. To find the area between two curves
. To find the area under a curve
. The average value of a curve
How to Determine the Area Under the Curve?
Let us assume the curve y=f(x) and its ordinates at the x-
axis be x=a and x=b. Now, we need to evaluate the area
bounded by the given curve and the ordinates given by
x=a and x=b.

12. The area under the curve can be assumed to be made up
of many vertical, extremely thin strips. Let us take a
random strip of height y and width dx as shown in the
figure given above whose area is given by dA.
The area dA of the strip can be given as y dx. Also, we
know that any point of the curve, y is represented as f(x).
This area of the strip is called an elementary area.
This strip is located somewhere between x=a and x=b,
between the x-axis and the curve. Now, if we need to find
the total area bounded by the curve and the x-axis,
between x=a and x=b, then it can be considered to be
made of an infinite number of such strips, starting from
x=a to x=b. In other words, adding the elementary areas
between the thin strips in the region PQRSP will give the
total area.
Using the same logic, if we want to calculate
the area under the curve x=g(y), y-axis
between the lines y=c and y=d, it will be
given by:

13. Let us consider an example, to understand the
concept in a better way.

15. Conclusion
In the exploration of integrals, we have delved into a fundamental
and versatile concept within the realm of calculus. The journey
through this project has illuminated the diverse applications of
integrals in various fields, ranging from geometry and physics to
economics and engineering. Here are the key takeaways:
Understanding the calculation of the area under curves has been a
central theme. The conceptualization of the area as an
accumulation of infinitesimally small strips, each contributing an
elementary area, has provided a clear framework for applying
definite integrals to find total bounded areas.
Area under curves:
Applications in Real-world Scenarios:
Integrals have proven to be indispensable in modeling and
solving real-world problems. Whether calculating the total
distance traveled, determining the center of mass, or finding the
volume of a solid of revolution, the applications are vast and
impactful.
In conclusion, our exploration of integrals has been a journey into
the heart of calculus, unraveling its significance in understanding
change, accumulation, and the geometry of curves. The power of
integrals extends beyond the confines of textbooks, finding its
place in the analysis of dynamic systems, the optimization of
processes, and the unraveling of the mysteries of the natural
world. As we conclude this project, the door to further
exploration of calculus and its myriad applications stands wide
open, inviting us to delve deeper into the mathematical wonders
that surround us.

16. Thank you