A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL INTEGRAL PROBLEMS

Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 4, December 2019
DOI : 10.5121/mathsj.2019.6401 1
A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3
AND SIMPSON’S 3/8 RULES OF NUMERICAL
INTEGRAL PROBLEMS.
Md. Jashim Uddin1
, Mir Md. Moheuddin2
and Md. Kowsher1
1
Dept. of Applied Mathematics, Noakhali Science and Technology University,
Noakhali-3814, Bangladesh.
2
Dept. of CSE (Mathematics), Atish Dipankar University of Science and
Technology (ADUST), Dhaka-1230, Bangladesh.
ABSTRACT
The main goal of this research is to give the complete conception about numerical integration including
Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of
Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules
demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to
determine the best method, as well as the results, are compared. It includes graphical comparisons
mentioning these methods graphically. After all, it is then emphasized that the among methods considered,
Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving
a definite integral.
KEYWORDS
Integration, Newton-Cotes formula, Trapezoidal method, Simpson’s 1/3 method, Simpson’s 3/8 method.
1. INTRODUCTION
Integration, which is a process of measuring the area plotted on a graph by a function as follows,
𝑏
𝐼 = ∫ 𝑓(𝑥)𝑑𝑥
𝑎
is the total value or summation of f(x) dx over the range from a to b. The system of estimating the
value of a definite integral from the approximate numerical values of the integrand, known as
numerical integration. A function of a single variable which is exerted in numerical integration is
called quadrature as well as expresses the area under the curve f(x). Besides, there are no
singularities of the integrand in the domain under the assumption and also numerical integration
comprises a boarding family of algorithms for the sake of counting the numerical values of a
definite integral. Nowadays, it is essential due to computers are too able to go through the
analytic manner of integration, even associating between analytical schemes and computer
processor.
In 1915, the term 'Numerical Integration' had first demonstrated in the publication of A Course in
Interpolation as well as Numeric Integration for the Mathematical Laboratory by David Gibb.
There are several application fields in numerical integration as like applied mathematics,
statistics, economics, and engineering, etc. Various methods are available in numerical
integration, for example, Quadrature methods, Gaussian integration, Monte-Carlo integration,
Adaptive Quadrature, and Euler-Maclaurin formula which are used to calculate those functions

2
𝑎
that are not integrated so easily. The various formula of numerical integrations is recounted in the
books of S.S. Sastry[11], R.L. Burden[12], J.H. Mathews[13] as well as numerous other authors.
J. Oliver [14] investigated the several processes of evaluation of definite integrals using higher-
order formula. Besides, Gerry Sozio [15] discussed a detailed summary of different techniques of
numerical integration. Using Bayesian methods, numerical integration is engaged in estimating
likelihoods and posterior distributions [16].
Moreover, the value of definite integral∫
𝑏
𝑦 𝑑𝑥which is enumerated by replacing the function y
using an interpolation formula and then integrated between a and b. In such a way, we can obtain
quadrature formula for which numerical values are acquainted as well. In many practical
circumstances, it is inevitable and more necessary than numerical differentiation.
In our working procedure, we have investigated and also compared with the existing some
Newton-Cotes methods such as the Trapezoidal rule, Simpson's 1/3 rule as well as Simpson's 3/8
rule to achieve the best results among them. Moreover, we demonstrated some sub-interval
randomly for determining the integral numerically and applied numerical examples to compare
our solutions with the exact value showing some condition graphically to obtain the effective
method which gives lesser error value among the mentioned methods.
A. PURPOSE AND MOTIVES OF THE STUDY
The main purpose of this study is to evaluate the method which is the best for solving the definite
function applying numerical methods. The objectives of the study are given below,
 Estimating the low error value of a solution, convergence as well as accurate results from the
other remaining methods.
 Comparing the existing methods for computing the appropriate method concerning the given
problems.
2. RELATED WORK
In the present era, numerical integration plays an extremely significant role in mathematics
affiliate, still, it is one of the branches joining the analytical calculations as well as computer
analysis. On the other hand, a large number of researchers have already been done comprehensive
research tasks with a view to modeling and promoting the several fields of numerical integration
for different objectives.
Besides, for instance, Ohta et al. [1] have compared various numerical integration to search out
the most effective method for the Kramers-Kronig transformation, applying the analytical
formula of the Kramers-Kronig transformation of a Lorentzian function as a reference. Also, they
compared their methods including the application of (1) Maclaurin's formula, (2) trapezium
formula, (3) Simpson's formula, and (4) successive double Fourier transform methods. In [2],
Siushansian, R. et al. demonstrated how the convolution integral arising in the electromagnetic
constitutive relation can be approximated by the trapezoidal rule of numerical integration as well
as implemented using a newly derived one-time-step recursion relation. Moreover, in their paper,
they have presented a comparison of different time-domain numerical techniques to model
material dispersion. However, Pennestrì et al. [3] gave and compared eight widespread
engineering friction force models, focused the attention on well-known friction models as well as
delivered a review and comparison based on numerical efficiency.

3
3. MATERIALS AND METHODS
Because of these tasks, the following methods had compared; Trapezoidal method, Simpson’s 1/3
method, and Simpson’s 3/8 method.
A. GENERAL QUADRATURE FORMULA
Let I = ∫
𝑏
𝑦 𝑑𝑥, where y=f(x).Let f(x) be given for certain equidistant values of x=𝑥 , 𝑥 +2h ,...,
𝑎 0 0
𝑥0+kh. Suppose 𝑦0,𝑦1,……,𝑦𝑘 are the entries corresponding to the arguments 𝑥0= a, 𝑥1= a+h,
𝑥2= a+2h ,...., 𝑥𝑘= a+kh = b respectively. Then we obtain,
∴ I = ∫
𝑏
𝑦𝑑𝑥 = ∫
𝑥0+𝑘ℎ
𝑦 dx
𝑎 𝑥0 𝑥
We know, u = 𝑥−𝑥0
ℎ
or x= 𝑥0 + uh ∴ dx = hdu
Limits: When x=𝑥0, then u=0
When x= 𝑥0+kh, then u = k
∴ I = ∫
𝑥0+𝑘ℎ
𝑦 𝑑𝑥 = ∫
𝑘
𝑦 h du
𝑥0 0 𝑥𝑜+𝑢ℎ
=ℎ ∫
𝑘
[
𝑦 + 𝑢
∆
𝑦 +
𝑢(𝑢−1)
∆
2
𝑦 +
𝑢(𝑢−1)(𝑢−2)
∆
3
𝑦 + ⋯ +
𝑢(𝑢−1)(𝑢−𝑛+1)
∆𝑘
𝑦 ]𝑑𝑢
0 0 0 2! 0 3! 0 𝑛! 0
What is more, in [4], Uilhoorn et al. attempted to search a fast and robust time integration solver
to obtain gas flow transients within the framework of particle filtering and investigated both stiff
and nonstiff solvers, namely embedded explicit Runge–Kutta (ERK) schemes. Bhonsale et al. [5]
basically presented a comparison between three different numerical solution strategies for
breakage population balance models and their results achieved for the fixed pivot technique,
moving pivot technique and the cell average technique. Furthermore, these approaches,
Concepcion Ausin, M. [6] compared various numerical integration producers and examined about
more advanced numerical integration procedures. In [7], Rajesh Kumar Sinha et al. have worked
to estimate an integrable polynomial discarding Taylor Series.
To solve Optimal Control Problem, Docquier, Q. et al. [8] explored the different dynamic
formulations and compared their performances and their focus had on minimal coordinates and
the derivation of the dynamics via the recursive methods for tree-like MBS (i.e., the so-called
Newton-Euler and Order-N recursive algorithms). In their paper, they introduced different
formulations and discussed their derivations. In [9], Parisi, V. et al. approach the classical,
Newtonian, gravitational N-body problem utilizing a new, original numerical integration method
and give the new algorithm, which is used to a set of sample cases of initial conditions in the
`intermediate' N regime (N=100). Yet Brands, B. et al [10] have tested the comparison of the
aforementioned hyper-reduction techniques focusing on accuracy and robustness, the well-known
DEIM is disapproved for their application as it suffers from serious robustness deficiencies.
Unlike these works, we discussed and investigated the most general one, namely the Newton-
Cotes methods involving the Trapezoidal, Simpson’s 1/3 and Simpson’s 3/8 rules. Several
procedures compared and endeavoured to display better methods with a few error values among
the existing methods, even to estimate the more proper values of definite integrals.

4
= ℎ[𝑘𝑦0 +
𝑘2
2
∆𝑦0 +(
𝑘3
3
𝑘2
− )
2
∆2𝑦0
2!
𝑘4
+ (
4
− 𝑘3 + 𝑘2)
∆3𝑦0
3!
𝑘5
+ (
5
3𝑘4
−
2
11𝑘3
+
3
− 3𝑘2)
∆4𝑦0
4!
+ ⋯ ]
This is the required Newton-Cotes method i.e, general quadrature formula. When k = 1, 2, 3......
then we obtain the Trapezoidal rule, Simpson’s 1/3 rule, Simpson’s 3/8 rule respectively. There
are some graphical examples of Newton-Cotes where the integrating function can be polynomials
for any order-for instance, (a) straight lines or (b) parabolas. The integral can be approximated in
one step or in a series of steps to develop accuracy as,
Figure. 1 Graphical examples of Newton-Cotes.
B. THE GENERAL FORMULA OF TRAPEZOIDAL RULE
In numerical analysis, the trapezoidal rule or method is a idea for approximating the definite
integral, the average of the left and right sums as well as usually imparts a better approximation
than either does individuallyThe basic idea of Trapezoidal rule graph is below .
𝑥𝑘
𝐼 = ∫ 𝑓(𝑥)𝑑𝑥
𝑥0
Figure. 2 Trapezoidal rule.
Also, we know from Newton-Cotes general quadrature formula that

5
1
𝑎
𝑏
0 1 0
I = ℎ [
𝑘
𝑦 +
𝑘2
∆
𝑦 + (
𝑘3
−
𝑘2
)
∆2𝑦0
+ (
𝑘4
− 𝑘3 + 𝑘2)
∆3𝑦0
+ (
𝑘5
−
3𝑘4
+
11𝑘3
− 3𝑘2)
∆4𝑦0
+
0 2 0
⋯ ]
3 2 2! 4 3! 5 2 3 4!
Now, putting k =1 in the above formula and neglecting the second and higher difference we get,
∫
𝑥0+ℎ
𝑦 𝑑𝑥= h
[
𝑦 + ∆𝑦 ]
𝑥0 0 2 0
= h[ 𝑦 +
1
(𝑦 − 𝑦 )]
2
=1 ℎ [( 𝑦 + 𝑦 )]
2 0 1
Similarly, ∫
𝑥0+2ℎ
y dx=
1
ℎ (𝑦 + 𝑦 )
𝑥0+ℎ 2 1 2
∫
𝑥0+𝑘ℎ 𝑦 𝑑
𝑥 =1 ℎ (𝑦 + 𝑦
⋯⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯⋯ ⋯
⋯⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯⋯ ⋯
)
𝑥0+(𝑘−1)ℎ 2 𝑘−1 𝑘
Adding these all integrals, we get,
∫
𝑥0+𝑘ℎ
𝑦 𝑑𝑥 =
ℎ [ 𝑦 + +2(𝑦 𝑦 + 𝑦 ........+𝑦 ) +𝑦 ]
𝑥0 2 0 1 2 3 𝑘−1 𝑘
This rule is acquainted as the trapezoidal rule.
C. THE GENERAL FORMULA OF SIMPSON’S ONE-THIRD RULE
In numerical integration, the Simpson’s 1/3 rule is a numerical scheme for discovering the
integral ∫
𝑏
𝑦 𝑑𝑥within some finite limits a and b. Simpson’s 1/3 rule approximates f(x) with a
polynomial of degree two p(x), i.e a parabola between the two limits a and b, and then searches
the integral of that bounded parabola which is applied to exhibit the approximate integral
∫𝑎
𝑦 𝑑𝑥. Besides, Simpson’s one-third rule is a tract of trapezoidal rule therein the integrand is
approximated through a second-order polynomial. The basic idea of Simpson’s one-three graph is
as follows:
Figure. 3 Simpson’s 1/3 rule.

6
Now, we know from Newton-Cotes general quadrature formula that
I = ℎ [
𝑘
𝑦 +
𝑘2
∆
𝑦 + (
𝑘3
−
𝑘2
)
∆2𝑦0
+ (
𝑘4
− 𝑘3 + 𝑘2)
∆3𝑦0
+ (
𝑘5
−
3𝑘4
+
11𝑘3
− 3𝑘2)
∆4𝑦0
+
0 2 0
⋯ ]
3 2 2! 4 3! 5 2 3 4!
Putting k =2 in the formula and neglecting the third and higher difference we get,
𝑥0+2ℎ 8
( −2)
∫ 𝑦 𝑑𝑥 = h [2𝑦0 + 2∆𝑦0+ 3
∆2𝑦0]
𝑥0
= h [
2
𝑦
2
+ 2(𝑦 − 𝑦 ) +1(𝑦 − 2𝑦 +𝑦 )]
0 1 0 3 2 1 0
=1 h (y + 4y + y )
3 0 1 2
Similarly,∫
𝑥0+4ℎ
𝑦 𝑑𝑥 =1 ℎ(𝑦 + 4𝑦 + 𝑦 )
𝑥0+2ℎ 3 2 3 4
⋯⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯⋯
⋯⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯⋯ ⋯
∫
𝑥0+𝑘ℎ 𝑦 𝑑𝑥 =1 ℎ(𝑦 + 4𝑦 +𝑦 )
𝑥0+(𝑘−2}ℎ 3
When k is even.
𝑘−2 𝑘−1 𝑘
Adding these all integrals, we obtain,
∫
𝑥0+2ℎ
𝑦 𝑑𝑥 +∫
𝑥0+4ℎ
𝑦 𝑑𝑥 + ⋯+ ∫
𝑥0+𝑘ℎ
𝑦 𝑑𝑥
𝑥0 𝑥0+2ℎ 𝑥0+(𝑘−2}ℎ
=1 ℎ [(𝑦 + 𝑦 )+ 4(𝑦 + 𝑦 + ⋯+𝑦 ) + 2
(
𝑦 + 𝑦 + ⋯+ 𝑦 )]
3 0 𝑘 1 3 𝑘−1 2 4 𝑘−2
Or, ∫
𝑥0+𝑘ℎ
𝑦 𝑑𝑥=
ℎ
[(y0 + yk) + 4(y1 + y3 + ..... + yk-1)+2(y2 + y4 + ... + yk-2)].
𝑥0 3
This formula is known as Simpson’s one-third rule. If the number of sub-divisions of the interval
is even then this method is only applied.
D. THE GENERAL FORMULA OF SIMPSON’S THIRD-EIGHT RULE
Simpson’s three-eight rule is a process for approximating a definite integral by evaluating the
integrand at finitely many points and based upon a cubic interpolation rather than a quadratic
interpolation. The different is Simpson’s 3/8 method applies a third-degree polynomial(cubic) to
calculate the curve. The basic idea of Simpson’s three eighth’s graph is as follows:

7
+( ) +( ]
Figure. 4 Simpson's 3/8 rule.
Further, we know from Newton-Cotes general quadrature formula that
I = ℎ [
𝑘
𝑦 +
𝑘2
∆
𝑦 + (
𝑘3
−
𝑘2
)
∆2𝑦0
+ (
𝑘4
− 𝑘3 + 𝑘2)
∆3𝑦0
+ (
𝑘5
−
3𝑘4
+
11𝑘3
− 3𝑘2)
∆4𝑦0
+
0 2 0
⋯ ]
3 2 2! 4 3! 5 2 3 4!
Putting k =3 in the formula and neglecting all differences above the third, we get,
∫
𝑥0+3ℎ
𝑦 𝑑𝑥 = ℎ [
3
𝑦 9 27 9 ∆2𝑦0 81 ∆3𝑦0
+ ∆𝑦 − − 27 + 9)
𝑥0
0 2 0 3 2 2! 4 3!
= h
9 9 8
[3𝑦0 + (𝑦1 − 𝑦0)+ (𝑦2 − 2𝑦1+𝑦0)+ (𝑦3 − 3𝑦2+3𝑦1−𝑦0)]
2 4 3
∫
𝑥0+3ℎ
𝑦 𝑑𝑥 =
3
ℎ(𝑦 + 3𝑦 +3𝑦 + 𝑦 )
𝑥0 8 0 1 2 3
Similarly, ∫
𝑥0+6ℎ
𝑦 𝑑𝑥 =
3
ℎ
(
𝑦 + 3𝑦 +
3
𝑦 + 𝑦 )
𝑥0+3ℎ 8 3 4 5 6
⋯⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯⋯ ⋯
⋯⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯⋯ ⋯
∫
𝑥0+𝑘ℎ 𝑦 𝑑𝑥 =3 ℎ(𝑦 + 3𝑦 +3𝑦 + 𝑦 )
𝑥0+(𝑘−3)ℎ 8 𝑘−3 𝑘−2 𝑘−1 𝑘
Adding these all integrals, we get,
∫
𝑥0+3ℎ
𝑦 𝑑𝑥 +∫
𝑥0+6ℎ
𝑦 𝑑𝑥 + ⋯ + ∫
𝑥0+𝑘ℎ 𝑦 𝑑𝑥
𝑥0 𝑥0+3ℎ 𝑥0+(𝑘−3)ℎ
=
3
h [(yo + yk) + 3(y1 + y2 + y4 + y5 + .... + yk-1) + 2(y3 + y6 + ... + yk-3)]
8
This formula is known as simpson’s three-eights rule.

8
0
4. RESULTS AND DISCUSSION
Problem-1: Suppose ∫
𝜋/2
sin(𝑥) 𝑑𝑥that is determined by using Simpson’s 1/3 rule, Simpson’s3/8
rule & Trapezoidal rule and interpreting the results by the three methods in this tasks, the results
of the methods are demonstrated in the following table as well as the comparison of the
approximate error is also given below.
K Exact
value
Simpson’s
1/3 rule
Error Simpson’s
3/8 rule
Error Trapezoid
al
rule
Error
1 1.000000 0.5235987 0.47640 0.589048 0.41095 0.785398 0.21460
2 1.000000 1.0022798 0.00228798 0.919304 0.08070 0.948059 0.05194
3 1.000000 0.8258986 0.17410 1.0010049 0.0010049 0.977048 0.02295
4 1.000000 1.0001345 0.0001345 0.900821 0.09918 0.987115 0.01288
5 1.000000 0.8953350 0.10466 0.961517 0.03848 0.991761 0.00824
6 1.000000 1.0000263 0.0000263 1.0000596 0.00006 1.251620 0.25162
7 1.000000 0.9252143 0.07479 0.943693 0.05631 0.995800 0.00420
8 1.000000 1.0000082 0.0000082 0.975634 0.02437 0.996785 0.0032
9 1.000000 0.9418275 0.05817 1.0000116 0.0000116 0.997460 0.00254
10 1.000000 1.0000033 0.0000033 0.960656 0.03934 0.997942 0.00206
11 1.000000 1.1428019 0.14280 0.982216 0.01778 1.140373 0.14037
12 1.000000 1.0000016 0.0000016 1.0000016 0.0000016 0.998571 0.00143
13 1.000000 1.1208316 0.12083 0.969758 0.03024 1.119173 0.11917
14 1.000000 1.00000088 0.00000088 0.986006 0.01399 0.998950 0.00105
15 1.000000 1.1047204 0.10472 1.0000015 0.0000015 1.103518 0.10352
16 1.000000 1.00000051 0.00000051 0.975437 0.02456 0.999196 0.00080
17 1.000000 1.0924001 0.09240 0.988467 0.01153 1.091491 0.09149
18 1.000000 1.00000032 0.00000032 1.00000032 0.00000032 1.086465 0.08647
19 1.000000 0.9724424 0.02756 0.979320 0.02068 0.999430 0.00057
20 1.000000 1.00000021 0.00000021 0.990193 0.00981 0.999485 0.00051
21 1.000000 0.9750668 0.02493 1.00000039 0.00000039 0.999533 0.00047
22 1.000000 1.00000014 0.00000014 0.982142 0.01786 1.070884 0.07088
23 1.000000 1.0682956 0.06830 0.991469 0.00853 1.067827 0.06783
24 1.000000 1.00000010 0.00000010 1.00000022 0.00000022 1.065022 0.06502
25 1.000000 0.9790561 0.02094 0.984287 0.01571 0.999670 0.00033
26 1.000000 1.000000074 0.000000074 0.992452 0.00755 1.060055 0.06006
27 1.000000 0.9806075 0.01939 1.00000014 0.00000014 0.999717 0.00028
28 1.000000 1.000000055 0.000000055 0.985971 0.01403 0.999737 0.00026
29 1.000000 0.9819449 0.01806 0.993232 0.00677 0.999755 0.00024
30 1.000000 1.000000041 0.000000041 1.000000094 0.000000094 0.9997715 0.00023
31 1.000000 0.9831097 0.01689 0.987329 0.01267 0.999786 0.00021
32 1.000000 1.000000032 0.000000032 0.993866 0.00613 0.999799 0.00020
33 1.000000 0.9841333 0.01587 1.000000064 0.000000064 0.999811 0.00019
34 1.000000 1.000000025 0.000000025 0.988448 0.01155 0.999822 0.00018
35 1.000000 0.9850400 0.01496 0.994391 0.00561 0.999832 0.00017
36 1.000000 1.000000020 0.000000020 1.000000020 0.000000020 1.043453 0.04345
37 1.000000 0.9858486 0.01415 0.989384 0.01062 0.999849 0.00015
38 1.000000 1.000000016 0.000000016 0.994834 0.00517 1.041176 0.04118
39 1.000000 0.9865744 0.01343 1.000000032 0.000000032 0.999864 0.00014
40 1.000000 1.000000013 0.000000013 0.990181 0.00982 0.999871 0.00013
Table 1: The results of the three methods.
Here K = 1 to 40 which is the number of subdivision of the interval of the integration.

9
0
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Comparison of the Approximate Error
Simpson 1/3 Rule
Simpson 3/8 Rule
Trapezoidal Rule
0 2 4 6 8 10 12 14 16 18 20
Subdivisions
Figure. 5 The approximate error is plotted against the number of subdivisions (1-20).
From the above table and comparison of the approximate error, we can claim that Simpson’s 1/3
rule gives the lesser error value among other methods when the condition of the subdivision is
only even, other methods impart less accuracy in this case as compared to other methods. As a
result, it is recommended strongly that Simpson's 1/3 is the most robust method for solving a
definite integral and very close to the exact value.
Similarly,
Problem-2: Let ∫
6
𝑒𝑥 𝑑𝑥that is calculated by applying Simpson’s 1/3 rule, Simpson’s 3/8rule &
Trapezoidal rule and interpreting the results by the three methods in this tasks, the results of the
methods are demonstrated in the following table as well as the comparison of the approximate
error is also given below.
K Exact
value
Simpson’s
1/3 rule
Error Simpson’s
3/8 rule
Error Trapezoidal
rule
Error
1 402.428 808.85758 406.42958 909.96478 507.536 1213.28638 810.8576
2 402.428 484.77094 82.34294 522.77107 120.3423 666.89980 264.471
3 402.428 362.12087 76.30713 442.79280 40.36402 528.40320 125.9744
4 402.428 411.29757 8.86957 370.21766 32.2111 475.19813 72.76934
5 402.428 331.66775 70.76025 398.33387 4.09492 449.59961 47.17082
6 402.428 404.42370 1.9957 406.48342 4.054635 435.41858 32.98979
7 402.428 334.8533 67.5747 355.88397 46.5448 1007.08899 604.6602
8 402.428 403.09146 0.66346 399.33387 3.09492 421.11813 18.68934
9 402.428 342.02113 60.4077 403.32695 0.898158 417.22431 14.79552
10 402.428 402.70657 0.27857 361.32310 41.1057 755.9862 353.5582
11 402.428 348.79228 53.6365 388.37344 14.0554 712.22136 309.7926
12 402.428 402.56447 0.13647 402.72548 0.29669 410.77800 8.34921
13 402.428 354.53572 47.8931 367.25854 35.1703 409.54726 7.118476
14 402.428 402.50259 0.07459 389.3363 13.0925 408.56964 6.140854
15 402.428 359.31136 43.1174 402.55283 0.124044 407.78025 5.351466
16 402.428 402.47227 0.04427 372.06195 30.3668 407.13373 4.70495
17 402.428 363.29057 39.1382 390.50270 11.9261 579.11662 176.6878
18 402.428 402.45603 0.02803 402.48929 0.060501 567.22460 164.7958
19 402.428 366.63459 35.7942 375.83332 26.5955 405.76753 3.338742
Error

10
20 402.428 402.44671 0.01871 391.58997 10.8388 547.64259 145.2138
21 402.428 369.47354 32.9552 402.46167 0.032882 539.48788 137.0591
22 402.428 402.44105 0.01305 378.82213 23.6067 532.19502 129.7662
23 402.428 371.90827 30.5205 392.54317 9.88562 404.70841 2.279624
24 402.428 402.43746 0.00946 402.44815 0.019362 404.52259 2.093807
25 402.428 374.01628 28.4125 381.23121 21.1976 404.35859 1.92981
26 402.428 402.43509 0.00709 393.36556 9.06323 404.21313 1.784347
27 402.428 375.85740 26.5714 402.44091 0.012125 404.08351 1.654728
28 402.428 402.43348 0.00548 383.20684 19.222 403.96752 1.538732
29 402.428 377.47817 24.9506 394.07416 8.35463 403.86330 1.434517
30 402.428 402.43235 0.00435 402.43676 0.007973 403.76932 1.340539
31 402.428 378.91522 23.5136 384.852820 17.576 490.10452 87.67573
32 402.428 402.43154 0.00354 394.68724 7.74155 403.60709 1.178304
33 402.428 380.19764 22.2312 402.43424 0.005454 484.20041 81.77162
34 402.428 402.43095 0.00295 386.24348 16.1853 403.47261 1.043827
35 402.428 381.34879 21.08 395.22094 7.20785 403.41385 0.985061
36 402.428 402.43051 0.00251 402.43264 0.003856 403.35991 0.931121
37 402.428 382.38764 20.0412 387.43295 14.9958 403.31028 0.881492
38 402.428 402.43017 0.00217 395.68863 6.74016 403.26451 0.835727
39 402.428 383.32968 19.042 402.43159 0.002802 403.22222 0.793436
40 402.428 402.42992 0.00192 388.46136 13.9674 403.18306 0.75506
Table 2: The results of the three methods.
Here K=1 to 40 which is the number of subdivision of the interval of the integration.
Comparison of the Approximate Error
900
800
700
600
500
400
300
200
100
0
0 2 4 6 8 10 12 14 16 18 20
Subdivisions
Figure. 6 The approximate error is plotted against the number of subdivisions (1-20).
From the above table and comparison of the approximate error, we can claim that Simpson’s 1/3
rule gives the lesser error value among other methods when the condition of the subdivision is
only even, other methods impart less accuracy in this case as compared to other methods. As a
result, it is recommended strongly that Simpson's 1/3 is the most robust method for solving a
definite integral and very close to the exact value.
Simpson 1/3 Rule
Simpson 3/8 Rule
Trapezoidal Rule
Error

11
sin(x)
A. VERIFICATION TO ACHIEVE THE BEST METHOD FOR PROBLEM-1 & PROBLEM-2
GRAPHICALLY
Firstly, we get the following graphical comparison for problem-1 when subinterval is 4.
Given graph of sin(x)
1
Graph of sin(x) using Trapezoidal rule
1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0
0 10 20 30 40 50 60 70 80 90
x-axis
0
0 10 20 30 40 50 60 70 80 90
x-axis
1
0.8
0.6
0.4
0.2
0
Graph of sin(x) using Simpsons one-third
0 10 20 30 40 50 60 70 80 90
x-axis
1
0.8
0.6
0.4
0.2
0
Graph of sin(x) using Simpsons third-eighths
Simpson 3/8 Rule
0 10 20 30 40 50 60 70 80 90
x-axis
Figure.7 Graphical comparisons.
Comparing the above graph,we see that Simpson’s 1/3 is a better method than others.
Similarly,
Secondly, we obtain the following graphical comparison for problem-2 when subinterval is 6.
400
300
200
100
0
Given graph of exp(x)
exp(x)
0 1 2 3 4 5 6
x-axis
400
300
200
100
0
Graph of exp(x) using Trapezoidal rule
0 1 2 3 4 5 6
x-axis
400
300
200
100
Graph of exp(x) using Simpsons one-third
400
300
200
100
Graph of exp(x) using Simpsons third-eighths
0
0 1 2 3 4 5 6
x-axis
0
0 1 2 3 4 5 6
x-axis
Figure. 8 Graphical comparisons.
Comparing the above graph,we see that Simpson’s 1/3 is a better method than others.
Trapezoidal Rule
Simpson 1/3 Rule
Trapezoidal Rule
y-axis
y-axis
y-axis
y-axis
y-axis
y-axis
y-axis
y-axis
Simpson 1/3 Rule Simpson 3/8 Rule

12
5. CONCLUSION
From the methods examined in our paper, we are capable of showing numerical integration for
finding the smallest error value by using the methods of Trapezoidal as well as Simpson’s 1/3,
Simpson’s 3/8 rules that we have discussed. In our paper tasks, we have tried to display some
examples as well as emphasized the condition for which Simpson’s one-third method is the best.
Consequently, we see that Simpson’s one-third rule gives the smallest error value among the rules
as well as formally it is the most effective and appropriate methods among the mentioned rules in
the case of even subdivision. s
ACKNOWLEDGEMENT
We are especially thankful to reveal our heartiest gratitude and sincerest liability to Md. Jashim
Uddin, Assistant Professor, Dept. of Applied Mathematics, Noakhali Science and Technology
University for imparting us the valuable suggestion and constant encouragement to work on this
research field.
REFERENCES
[1] Ohta, Koji, and Hatsuo Ishida. "Comparison among several numerical integration methods for
Kramers-Kronig transformation." Applied Spectroscopy 42.6 (1988): 952-957.
[2] Siushansian, R., & LoVetri, J. (1995). A comparison of numerical techniques for modeling
electromagnetic dispersive media. IEEE Microwave and Guided Wave Letters, 5(12), 426-428.
[3] Pennestrì, Ettore, Valerio Rossi, Pietro Salvini, and Pier Paolo Valentini. "Review and comparison of
dry friction force models." Nonlinear dynamics 83, no. 4 (2016): 1785-1801.
[4] Uilhoorn, Ferdinand Evert. "A comparison of numerical integration schemes for particle filter-based
estimation of gas flow dynamics." Physica Scripta 93, no. 12 (2018): 125001.
[5] Bhonsale, S. S., Telen, D., Stokbroekx, B., & Van Impe, J. (2019). Comparison of numerical solution
strategies for population balance model of continuous cone mill. Powder Technology.
[6] Concepcion Ausin, M. (2007) an introduction to quadrature and other numerical integration
techniques, Encyclopedia of Statistics in Quality as well as reliability. Chichester, England.
[7] Rajesh Kumar Sinha, Rakesh Kumar, 2010, Numerical method for evaluating the integrable function
on a finite interval, International Journal of Engineering Science and Technoligy. Vlo-2(6).
[8] Docquier, Q., Brüls, O., & Fisette, P. (2019). Comparison and Analysis of Multibody Dynamics
Formalisms for Solving Optimal Control Problem. In IUTAM Symposium on Intelligent Multibody
Systems–Dynamics, Control, Simulation (pp. 55-77). Springer, Cham.
[9] Parisi, V., & Capuzzo-Dolcetta, R. (2019). A New Method to Integrate Newtonian N-Body
Dynamics. arXiv preprint arXiv:1901.02856.
[10] Brands, B., Davydov, D., Mergheim, J., & Steinmann, P. (2019). Reduced-Order Modelling and
Homogenisation in Magneto-Mechanics: A Numerical Comparison of Established Hyper-Reduction
Methods. Mathematical and Computational Applications, 24(1), 20.
[11] S.S Sastry, 2007, Introductory Method of Numerical Analysis, Fourth Edition, Prentice-hall of India
Private Limited.

13
[12] Richard L. Burden, 2007, Numerical Analysis, Seven Edition, International Thomson Publishing
Company.
[13] Jonh H. Mathew, 2000, Numerical Method for Mathematics, science and Engineering, Second
Edition, Prentice Hall of India Private Limited.
[14] J. Oliver, 1971, The evaluation of definite integrals using high-order formulae, The Computer
Journal, Vol-14(3).
[15] Gerry Sozio, 2009, Numerical Integration, Australian Senior Mathematics Journal, Vol-23(1).
[16] Evans M, Swartz T. Methods for approximating integrals in statistics with special emphasis on
Bayesian integration problems. Statistical Science. 19995;10:254-272.

A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL INTEGRAL PROBLEMS

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A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL INTEGRAL PROBLEMS