CONTENTS

1. Introduction 1
2. AIM 2
3. Theory 2
4. Requirements 9
5. Procedure 9
6. Observations 10
7. Result 11
8. Applications 11
9. Bibliography 12
Introduction
Integrals are fundamental tools in calculus, used to
calculate areas, volumes, and accumulations. Their history
traces back to ancient civilizations like Egypt and Greece,
where mathematicians such as Archimedes developed early
methods to find areas and volumes, laying the groundwork
for modern integration.

At its core, integration can be understood as a limit of

sums. This idea, known as the Riemann sum, forms the
foundation of the integral, bridging the gap between
discrete and continuous quantities.

The formal concept of integration emerged in the 17th

century with the advent of calculus, pioneered by Isaac
Newton and Gottfried Wilhelm Leibniz. They linked
integration with differentiation through the Fundamental
Theorem of Calculus, revolutionizing mathematics.

Today, integrals are indispensable across fields. They are

used in geometry for areas and volumes, physics for
mechanics and electromagnetism, engineering for structural
analysis, economics for growth models, and computer
science for simulations.
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From ancient methods to modern applications, integrals
remain a powerful tool for understanding and solving real-
world problems.

AIM
To evaluate definite integral:
As a limit of sum and verify it
by actual integration

Theory
Mathematically, an integral is represented by the
integral sign ∫, followed by the function to be integrated
(called the integrand) and a differential dx, which
signifies the variable of integration.
This integral is an indefinite
integral if there are no limits, which
are numbers written on the top
and bottom of the integral sign ∫,
but if there are limits, it is called
definite integration.
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Indefinite integration involves finding a general function
whose derivative is equal to the given integrand. The
result is a family of functions represented with an
arbitrary constant C, known as the constant of
integration.

This further solidifies and explains the connection

between integration and differentiation, since in
indefinite integration we are finding the antiderivative,
also called primitive of a function.

Since in indefinite integration, we are finding

antiderivative, it also follows that the derivative of the
family of functions you get from indefinite integration will
be the function itself. And this is what is said in the first
fundamental theorum of calculus.

First Fundamental Theorum Of

Calculus
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Definite integration calculates the net accumulation or
total value of a quantity over a specific interval [a,b]. It
results in a numerical value rather than a general
function.

If F(x) is the antiderivative of f(x), then the definite

integral value of f(x) in the interval [a,b] can be given by
F(b) - F(a) (there is no constant of integration here).
And this is called the second fundamental theorum of
calculus.

Second Fundamental Theorum

Of Calculus

This integral can be calculated in many ways algebraically.

There are some basic rules of integration that are derived as
being the reverse of the rules of differentiation like:

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There are also other methods of integrating like by using
partial fractions, trigonometric substitutions, integration by
parts, etc.

The definite integral can also be defined as a limit of sums,

which we will be verifying in this experiment.
Consider an increasing function as such given below.
Definite integral from 0 to 1 of this function f(x) will be the
area under the curve, as highlighted below.

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Now we shall try to estimate this area under the curve by
using 4 rectangles of width 0.25 units each and height as
f(a) for each a from {0.25, 0.5, 0.75, 1}. This area can be
seen in diagram.

This area we have estimated is equal to:

Area = 0.25 * f(0.25) + 0.25 * f(0.5) + 0.25 * f(0.75) +
0.25* f(1)
Area = 1/4 * (f(0.25)+f(0.5) + f(0.75) + f(1))

(Area of each rectangle = width (0.25) * height)

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Here is the excess area, or the error in our estimation of
area highlighted with red in our diagram.

Since there is a lot of error, let us try to see what happens

when we increase the number of rectangles we are finding
area of to be 10.

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We can clearly see that the error has greatly reduced. We
can now tell that to make an even better estimation, we
would have to draw even more rectangles to get better
precision.

As we increase the number of rectangles to let’s say n, then

the width of each rectangle would become 1/n

The height of the kth rectangle would be f(k/n)

And area of all rectangles would thus be:

Now as we increase number of rectangles, the error will

keep reducing.
We can also then say that when the limit of number of
rectangles or n approaches infinity, then the error in
estimation of area tends to 0 OR in other words, the sum of
areas of all rectangles will tend to the area under the curve.

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Similar logic can be applied to convert any definite integral
into a limit of sums, and that is the principle we will use in
this experiment.

Requirements
Cardboard, white drawing sheet, wires (colored), scale,
pencils, graph paper, glue, nails

Procedure
1) Take a cardboard of 30 x 200 cm and paste a white
drawing sheet on it

2) Now paste a graph paper on white drawing sheet and

draw two perpendicular lines representing x and y axis

3) Draw a quadrant of a circle with O(0,0) as center and

radius 1 unit (10cm) in the 1st quadrant. This represents
the graph of

4) Let quadrant cut x axis at A and y axis at B

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5) Divide OA into 10 equal parts at points named A1,
A2,..., A9

6) From these points, draw vertical lines meeting the curve

at B1, B2,.... B9

7) Measure lengths A1B1, A2B2, .... A9B9

Observation
1) Lengths: A1B1 = 9.95, A2B2 = 9.79, A3B3 = 9.53, A4B4
= 9.16, A5B5=8.66, A6B6= 8.00, A7B7=7.14, A8B8=6.00,
A9B9= 4.35
2)All the widths are 1cm

3) Sum of areas of all rectangles = 1(A1B1+A2B2+...+A9B9)

= 7.26 approximately.

4) Area under the curve using integration:

The area will be 10 times of
(since our scaling is by 10)

5) Error = 7.85-7.26 = 0.59

Error% = 0.59/7.85 * 100 = 7.51%

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 Result
Using by estimation using sum of areas of rectangles, we get
a result that is very close to the actual area under the curve.
Our error is only about 7.51% which is quite good.

Applications
Integrals have numerous applications across various fields.
In geometry, they calculate areas under curves and volumes
of solids by summing infinitesimal slices.

In physics, they model quantities like work done, where the

total force applied over a distance is the sum of infinitely
small force elements. Similarly, integration determines the
center of mass by summing up distributed mass elements.

In economics, integrals are used to compute total cost or

revenue by summing marginal contributions over time or
quantity.

In computer science and data analysis, numerical

integration methods approximate these sums for complex or
discrete datasets, bridging continuous and discrete systems.

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 Bibliography

NCERT Maths
Together with Maths Lab Manual
https://en.wikipedia.org/wiki/History_of_calculus
https://en.wikipedia.org/wiki/Integral
http://5010.mathed.usu.edu/Fall2013/EVanfleet/Integrati
on%20History.html
https://www.desmos.com/calculator
https://www.cuemath.com/calculus/fundamental-
theorem-of-calculus/
https://www.tutorialspoint.com/whiteboard.htm

Images:
https://www.desmos.com/calculator
https://calculator-
online.net/images/Artboard_%E2%80%93_25_165771001226
0.webp
https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_fil
es/ftc-1-and-ftc-2-1623911034.png
https://lh5.googleusercontent.com/proxy/bWDwPhUat2P-
4ZsUR6UcCp3jpITv9N4uZyagMRz-
BnYacwaYIUBQVEiZ3CYD4ZeBK073YC_yBRHp24NpkCHmFV-
acQj7YiezveMvHuEi5k7F5V5RIqdayFul32CCOu1xEGWSZJrGwjhQUSN
h7tZuUoKBFwEOeZe0qxnS

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