Integration as a Limit of Sum
Shreesh Swaraj
Roll no. IIT2019044, ID- iit2019044@iiita.ac.in
I Semester, BTech., Department of Information Technology
Indian Institute of Information Technology, Allahabad
Abstract- In this report I have discussed a Now, let us understand what does integrating
method to find the integral of a function f(x)=x2 would mean. Integrating the given
without using the standard formula. I have also equation with respect to x would mean finding
verified the result using the standard formula. the area made by the graph with x axis.
Keywords- Integration as a limit of sum. Now let us find the area of the given curve f(x)
=x2 made with x axis from x=a=0 to x=b=5. The
I. Introduction
technique that I have used is that we can divide
Integration, in simple words, is the summing up the graph into small n rectangular areas of small
of small elements. The mathematics behind width h as shown with h = (b-a)/n.
integration was discovered by Sir Isaac Newton.
It comes out to be very useful in mathematics
when finding area of any curve made with the
axes. Not only in mathematics but it finds many
important applications in physics and chemistry.
There are two type of integration that we can
perform, namely, definite and indefinite. While
performing indefinite integration provides us a
general formula to find the area under a certain
curve, definite integration provides us the value
of area made by the curve. The concept of
integration has helped people to discover the
unknown.
II. Proposed Technique
Graph: f(x)=x2 Fig: Graph of f(x) =x2 divided into small
120 rectagles.
Therefore the sum of areas of rectangles can be
100
given by
80
Sn = h[f(a) + f(a+h) + f(a+2h) +…..+ f(a+nh)] -(1)
60 We can say that Sn < area(OABO) -(2)
40 Now upon reducing the width of the rectangles
i.e. h→0, n→∞ and we can say that Sn
20 approaches a limiting value equal to the
area(OBAO).
0
0 5 10 15
Integration as a Limit of Sum
Shreesh Swaraj
Roll no. IIT2019044, ID- iit2019044@iiita.ac.in
I Semester, BTech., Department of Information Technology
Indian Institute of Information Technology, Allahabad
𝑏 STEP 3: Write the numeric limit for which you
lim 𝑆 = area of the region OABO = ∫𝑎 𝑓(𝑥)𝑑𝑥 –
𝑛→∞
want to find the algebraic area made by the
(3)
graph with the axis.
We can replace h in eq. (1) with h = (b-a)/n to
STEP 4: Type 1 if you want to find the algebraic
solve the integral as given in eq. 3.
sum of area else type 0 if you want to find the
This method to calculate integral is known as positive sum of the areas.
integral as a limit of sum.
STEP 5: If you want to perform indefinite
Now, let us evaluate the integral of f(x) =x2 from integration then skip step 3.
x=0 to x=5.
V. Conclusion
We have, a=0, b=5, h= (5-0)/n = 5/n
From this we can conclude that an integration
In = (5/n) [ 02 + (5/n)2 + (5.2/n)2 + (5.3/n)2 + ……. + can be performed without using the standard
(5.n/n)2 ] formulas and we can evaluate it by limiting the
sum.
Now taking (5/n)2 common we get,
In = (5/n) (5/n)2 [1 + 22 +32 + ……. + n2 ]
VI. References
Since sum of the square of first n natural
i. https://en.wikipedia.org/wiki/Integral
numbers is given by n(n+1)(2n+1)/6] we get,
ii. Chapter 7, Class XII NCERT Part 2
In = (5/n)3 [n(n+1)(2n+1)/6]
Now as n→∞, taking n common and eliminating
it we get,
In = 125/3
III. Verification using formula
From the standard formula we know that
2
∫ 𝑥 𝑑𝑥 = 𝑥 3 /3. So putting the limit from x=0 to
5 53
x=5 we get∫0 𝑥 2 𝑑𝑥 = 3
= 125/3.
IV. Algorithm
STEP 1: Write the equation of the curve which
needs to be integrated.
STEP2: Write the name of the axis (i.e. X or Y)
with which you want to find the area enclosed by
the graph.
