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Mathematics Education Journals 
Vol. 4 No. 1 February 2020 
 

 

ISSN : 2579-5724   
ISSN : 2579-5260 (Online) 

http://ejournal.umm.ac.id/index.php/MEJ 

 

 

 

Mathematical Construction of Definite Integral Concepts by Using GeoGebra 

Widiya Astuti Alam Sur 

 

Program Studi Akutansi, Politeknik Negeri Tanah Laut 
Jl Ahmad Yani Km. 06, Kec.Pelaihrai, Kab.Tanah Laut, Kalimantan Selatan 

e-mail: widiyasur@politala.ac.id 
 

Abstract 

The need to visualize concepts and support the materials in Mathematics learning by 
forming the pictures or by using existing draw, becoming what is needed by the 
students in improving their mathematical development and understanding. GeoGebra 
can combine dynamic visualization of geometry and the results of mathematical 
calculations simultaneously. The author examines the definite integral concept 
material of course, to then be explored with GeoGebra so that it is easily understood 
by students. The author studies the procedures for using GeoGebra, studies and 
constructs the concept of definite integral concepts, into a simulation and 
visualization in the form of GeoGebra files, which are ready to be used in the learning 
of definite integral concepts.. 

Ke ywords : GeoGebra; definite integral; mathematical construction. 
 

INTRODUCTION 

The use of technology in learning mathematics is not a new thing to use. At present, 

there are many Mathematics software specifically designed to facilitate the learning of 
Mathematics, both paid and which can be obtained free from the internet. Computer 
Algebra Systems (such as Derive, Mathematica, Maple or MuPAD), Dynamic Geometry 

Software (such as Geometer's Sketchpad or Cabri Geometry), Matlab, Autograph, and 
GeoGebra are some examples of software that are widely used in Mathematics learning. 

There is a need to visualize Math concepts and materials by imagining, shaping 

the images (manually or with the help of technology) or by using existing images, which 
is the reason why in Mathematics, we need media that can help the learning process. 

Furthermore, to gain a deeper understanding, visualization must be represented. Students 
must learn how mathematical ideas are represented by symbols, numbers, and graphics, 

as a form of understanding of mathematical concepts (Caligaris, 2014). 

For the reasons mentioned above, it is very necessary for a teacher's creativity in 
choosing and using appropriate learning media to bridge the mathematical abilities of their 

students. Along with the development of technology, there are many choices of media for 
learning mathematics that can be used for this. One of the software that can be obtained 

free from the internet is GeoGebra. GeoGebra is a dynamic mathematical software that 
can be used at various levels of education, from elementary school level to university level.  

The GeoGebra application combines geometry, algebra, spreadsheets, graphs, 

statistics, and calculus, in one package that is easy to use. This makes GeoGebra have 
quite a several users spread in almost every country throughout the world. Nowadays, 

GeoGebra has become a provider of software leading mathematical that supports the 



 

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advancement of science, technology, engineering, and mathematics education, as well as 

providing innovation in teaching and learning processes throughout the world. In this 
study, the application of GeoGebra 5, which is a new edition of GeoGebra, is used because 

it is easily obtained, and its application can reach various domains of Mathematics. 

Besides being able to visualize mathematical concepts that are analytic 
geometrically, GeoGebra is also able to perform mathematical and statistical calculations 

and represent them. To facilitate users, GeoGebra facilitates users to have an account and 
community on the GeoGebra website, so it is very easy to share and access Mathematics 

material that has been created by users in various countries. The user just has to choose 
the type of material and the desired material and can modify it according to what is needed. 
All of which are available completely and free of charge on the official GeoGebra 

website(Caligaris, 2014). 

Calculus is one of the mathematical materials that requires an understanding of 

concepts for students who study it, given the application and use of calculus that is very 
much needed in various branches of science (Serhan, 2015). However, the concept of 
Integral is certainly also difficult to understand and requires visualization to be able to 

imagine it, so it is appropriate if the writer chooses the topic of Integral and its application, 
to design simulations and learning media  

The aim of this research is to construct the Concept of Definite Integral, as well as 

design visualization and simulation of its material using GeoGebra. The number of errors 
in interpreting the meaning of Integral is certainly good among high school students or 

college students, making the writer feel the need to visualize the concept of the Integral of 
course. Of course by exploring and then designing simulation material in the form of 
dynamic learning media with GeoGebra. This article will present the results of the 

conceptual integration of Integral concepts with GeoGebra and visualization of concepts 
that are dynamic, interesting, easy to understand and provide a complete picture for 

understanding Integral concepts. 

RESEARCH METHOD 

The method of this article is literature studies. Data and documents obtained from 

various sources are compiled, analyzed, then the material produced is constructed and then 
visualized and simulated the material using GeoGebra. 

RESULT AND DISCUSSION  

The Overview of GeoGebra 

GeoGebra is a computer application program that combines geometry, algebra, 

tables, graphs, statistics, and calculus in an easy-to-use package so that it can be utilized 

in learning mathematics at various levels of education. GeoGebra is a dynamic program 

that can visualize or demonstrate and construct mathematical concepts, into various forms, 

types, styles for all levels of mathematical material. Besides being able to be obtained for 

free from the internet, GeoGebra is also easy to use and learn. There are many examples 

of material and tutorials that can be accessed on the official GeoGebra page. The files that 

are processed in GeoGebra can be saved in ".ggb" format, and or in the form of dynamic 



 

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web pages. GeoGebra can also produce ".png" or animated ".gif" files with quality 

illustrations. 

GeoGebra has received various educational software awards in Europe and the 

USA. Developed since 2001 by Markus Hohenwarter, a mathematician and Professor of 

Mathematics in Austria, who examines the special use of technology in mathematics 

education. GeoGebra continues to develop until now. Its inventors and designers continue 

to improve and add to the shortcomings of this GeoGebra program. The latest development 

is, GeoGebra 5., which can display images in 3-dimensional shapes. Aside from being a 

computer application, GeoGebra also has a special version for tablet or android users. It 

can even be used online without having to install the application first on a PC / Laptop. [3] 

With the development of applications and features on GeoGebra, its users are also 

expanding in various countries, ranging from students, students, teachers, and lecturers. 

Each user can create their account on the GeoGebra page and can join various GeoGebra 

user communities around the world. Users and owners of GeoGebra accounts can easily 

share and obtain a variety of materials or mathematical material in the form of GeoGebra, 

which can be downloaded and developed according to their learning needs. GeoGebra is 

formed on the Cartesian coordinate system and can accept geometrical commands 

(drawing lines through two points, forming curved fields, algebraic commands (drawing 

curves with given equations). Some of the advantages of GeoGebra compared to other 

mathematical software are (1) Can produce paintings - Geometry paintings quickly and 

thoroughly, even complex ones (2) The ability to represent algebra, two-dimensional 

geometry, and 3-dimensional geometry at once so that changes in given equations or 

changes in images in Geometry views will also change the shape of equations in Algebra 

display and position on the 3D display. (3) The existence of animation facilities and 

manipulation movements can provide visual experience in understanding the concept of 

geometry (4) Make it easy to investigate or show the properties that apply to a geometry 

object, (5) Can be used to make into learning material acts as a web page, and can be 

directly distributed to the official GeoGebra web site, www.GeoGebra.org. 

With these advantages, a teacher can guide their students to know the relationship 

between an algebraic form and geometry. Teachers can first design demonstrations for 

understanding students' concepts, and students can directly work from the materials 

needed and can see visually the meaning of the concepts being taught, thus helping them 

to be able to imagine the analytical meaning of a material being taught. 

When you open the GeoGebra application, Graph and Algebra displays and 

GeoGebra's standard menus will appear. Worksheets for displaying applications that have 

been created on GeoGebra are called applets. On the right side, there is an arrow which if 

clicked will bring up a box in the GeoGebra Math Application dialog. The dialog box 

provides display options or screen forms to be displayed, according to user needs. Where 

each view has its Toolbar which contains a selection of tools and commands that allow 



 

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users to create dynamic constructions in representing mathematical objects. There are 6 

views on GeoGebra, which can be selected based on user needs, namely: 

1) Graphing Calculator 
It is an Algebraic and Graphical display, as a standard display on GeoGebra. The left 

side, which is the Algebra display which is a place to display the algebraic object of 
the input object or equation, while the graph display will display the object 

geometrically 
2) Computer Algebra System (CAS) 

It is a computer algebraic system display that allows users to do mathematical 

calculations for symbolic calculations. The CAS display consists of lines that each 
have an input. 

3) Geometry 
Is a graphic display that only displays the geometric shapes of objects/equations that 
are inputted 

4) 3D Grapher  
Almost the same as Graphing Calculator, it's just that the graph shown is a 3 -

dimensional graph, with the left side is an Algebraic view. 
5) Spreadsheet 

It is a display form of a number processor table consisting of rows and columns. In 

this view, matrices, tables and other mathematical objects can be made that contain 
columns and rows. Not only numbers, but all types of mathematical objects can also 

be inputted into spreadsheet cells, such as point coordinates, functions, and 
mathematical calculation commands supported by GeoGebra. If possible, GeoGebra 
will immediately display a graphical representation of the object entered in the 

spreadsheet cell. 
6) Probability  

Is a display of statistical forms, which displays statistical distribution forms and 
statistical test calculations. 
(White, 2016). 

The following image is the home page of  GeoGebra worksheet. 

 

Figure 1 Home Page of  GeoGebra Worksheet 



 

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Visualization of Definite Integral Concepts with GeoGebra 

After knowing and exploring some basic material that is important to know in the 

understanding integral, of course, the following is given the construction of the 

presentation of the material of the concept of Integral Naturally by visualizing it on 
GeoGebra. Two things that become the basic idea of defining and constructing Integral of 
course are the Mileage and Space Area under the curve as follows : 

a. The Distance of Moving Objects 
In addition to the area under the curve, the integral idea is of course also based on the 
distance and speed of an object. Departing from real problems, the integral idea is 
certainly based on observing the distance of an object that moves (moves) within a 

certain time interval. If an object moves with a fixed speed at a certain time interval, 
then the total distance traveled can be calculated using a formula that has been known, 

namely  
Distance = speed × time 

For example, a train travels at a constant speed of 60km/ hour. Then the total distance 
traveled by train between 02:00 and 06:00 is 60 x 4 = 240 km. Calculation of the train's 
mileage, if visualized in the form of a speed function graph, can be seen in Figure 2 

(i). Because distance can be obtained by multiplying speed and time interval, then the 
total distance traveled is nothing but the area of the rectangle formed from the velocity 

function at that time interval. With the length of the interval as the length of the 
rectangle and the velocity value as the width of the rectangle. 
Calculation of the total distance in the interval of time taken by multiplying the speed 

and time used is if the object's speed remains (constant) 60 km/hour. But in reality, 
objects that move like the train, it is not possible to move at a constant speed within a 

certain time interval. There will be times when objects move slowly or quickly. If the 
velocity of the object changes within a certain time interval, then the visualization in 
the case of the train could be described as a function of speed in Figure 2(ii) 

 

Figure 2 Visualization of the Distance of Moving Objects with GeoGebra 

Figure 2 (ii) shows that the distance of the train is the same as the product of time 

inteval with the speed of the train. Therefore simply multiplying the length and width 
of the rectangle formed. However, in Figure 2 (ii) the shaded part is no longer 

rectangular. So to calculate it also requires a different approach. The distance traveled 



 

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can be calculated with the approach that best approaches the actual mileage. The 

approach taken is to partition the area into small rectangles, then add up the total area 
of the resulting rectangles. 

The time interval [2,6] can be partitioned into a number of very small sub-intervals, so 
that each of sub-intervals will show a constant velocity, because of very small time 
intervals. Geometrically, the subinterval division will produce small rectangles, along 

the area formed under the given speed function curve. 

 

Figure 3 Visualization of Interval Time Partition of Moving Object 

For the case of a railroad in Figure 3, the form of time interval division becomes area 

of the rectangles from this partition, then added up, so the results will approach the 
actual mileage. In this case, the area under the curve. 
If the train problem above is abstracted, then let's say an object moves continuously in 

the time interval [a, b], with a non-constant speed. Time lapse [a; b] can be partitioned 
into smaller time intervals, namely: 

[𝑡0, 𝑡1],[𝑡1, 𝑡2], . . . [𝑡{𝑛−1}, 𝑡𝑛] with 𝑎� = 𝑡0 < 𝑡1 <.. .< 𝑡𝑛 = �𝑏 

for each subintervals [𝑡𝑖−1, 𝑡𝑖], the object reaches 𝑀𝑖 maximum speed and reaches 
𝑚𝑖�minimum speed. So, if the object moves constantly at a minimum speed, the object 
will travel a distance of 𝑚𝑖∆𝑡𝑖 unit of distance. Conversely, if the object is constantly 
moving at maximum speed, then the object will travel a distance 𝑀𝑖∆𝑡𝑖 unit of 
distance. Meanwhile, the actual distance, denoted 𝑠𝑖, 𝑡ℎ𝑒 distance must be located 
between the maximum distance and the minimum distance, to obtain: 

𝑚𝑖∆𝑡𝑖 ≤ 𝑠𝑖 ≤ 𝑀𝑖∆𝑡𝑖 

The total distance traveled along the time interval [a,b] called s, is the sum of the 

distance traveled at each sub interval [𝑡{𝑖−1}��,𝑡𝑖],�then: 

𝑠 = 𝑠1 + 𝑠2 + ⋯+ 𝑠𝑛 

𝑚1∆𝑡1 ≤ 𝑠1 ≤ 𝑀1�∆𝑡1 

𝑚2∆𝑡2 ≤ 𝑠2 ≤ 𝑀2�∆𝑡2 

⋮ 

𝑚𝑛∆𝑡𝑛 ≤ 𝑠𝑛 ≤ 𝑀𝑛�∆𝑡𝑛 



 

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which in turn, add up the obtained inequality: 

 
𝑚1∆𝑡1 + 𝑚2∆𝑡2 + ⋯ + 𝑚𝑛∆𝑡𝑛 ≤ 𝑠 ≤ 𝑀1�∆𝑡1 + 𝑀2�∆𝑡2 + ⋯ + 𝑀𝑛�∆𝑡𝑛 (1) 

The sum of 𝑚1∆𝑡1 + 𝑚2∆𝑡2+�.. . +𝑚𝑛∆𝑡𝑛�𝑖𝑠 called the sum down for the velocity 
function and 𝑀1�∆𝑡1 + 𝑀2�∆𝑡2+�.. . +𝑀𝑛�∆𝑡𝑛 called the sum of the velocity functions. 
The inequality above shows that 𝑠 must be greater than or equal to each bottom sum, 
and smaller or equal to each top sum. As with the area, there is exactly one number 
that satisfies the inequality, and that number is the total distance traveled. 

 
b. Area Under The Curve 

Area under the curve becomes the basic idea of the formation of an understanding of 
Definite integral. Taking into account the following figure, suppose that A is the area 
under the curve 𝑦� = �𝑓�(𝑥) and above the axis 𝑥 − which is between the vertical lines 
𝑥� = �𝑎 and 𝑥� = �𝑏, then the area can be estimated by dividing region 𝐴 into 𝑛� 
subregions 𝐴1, �𝐴2, . . . , �𝐴𝑛 as in the following figure, presented on the GeoGebra 
worksheet as: 

 
Figure 4 Estimation of Area Under the Curve   𝑦 = 𝑓(𝑥) 

Because 𝑓�𝑖𝑠 continuous at intervals [𝑎,𝑏], then 𝑓�𝑖𝑠 continuous at each subinterval 

[𝑥{𝑖−1} ,𝑥𝑖] on P. This means, 𝑓 reaches the maximum and minimum at the points that 

exist in the subinterval. Therefore, there are numbers 𝑙𝑖 and 𝑢𝑖  in [𝑥{𝑖−1} ,𝑥𝑖] such that 

holds: 

𝑓(𝑙𝑖) ≤ 𝑓�(𝑥)�≤ �𝑓(𝑢𝑖)��𝑤𝑖𝑡ℎ��𝑥{𝑖−1} ≤ 𝑥� ≤ 𝑥𝑖 

Denoted 𝑀𝑖, in this case as 𝑓(𝑢𝑖) is the maximum value 𝑓 in the subinterval [𝑥{𝑖−1} ,𝑥𝑖] 

, and 𝑚𝑖in this case as 𝑓(𝑙𝑖) is the minimum value 𝑓 in the subinterval [𝑥{𝑖−1} ,𝑥𝑖]. If 

taking any subinterval to-i from the interval [a,b] the function 𝑓 of the following figure, 
it can be formed rectangle-rectangle 𝑟𝑖 and 𝑅𝑖 as shown below: 

 
Figure 5 Rectangle Area of Maksimum and Minimun Values of a Function 

 



 

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According to Figure 5, rectangular area 𝑟𝑖 and 𝑅𝑖 are formed on subintervals 
[𝑥𝑖,𝑥{𝑖−1}], which is then compared to the area 𝐴𝑖�at the same subinterval. It is clearly 

known if 𝑟𝑖 ⊆ 𝐴𝑖 ⊆ 𝑅𝑖, so that it is obtained: 
Area of 𝑟𝑖 ≤ Area of area 𝐴𝑖 ≤ Area of area 𝑅𝑖 
Because the area of a quadrilateral is a product of its length and width, we obtain  

𝑚_𝑖(𝑥𝑖 − 𝑥{𝑖−1}) ≤ �𝐴𝑟𝑒𝑎�𝑎𝑟𝑒𝑎�𝐴𝑖 �≤�𝑀𝑖(𝑥𝑖 − 𝑥{𝑖−1}) 

With △ 𝑥𝑖 = 𝑥𝑖 − 𝑥{𝑖−1}, we get:  

𝑚𝑖 △ 𝑥𝑖�𝐴𝑟𝑒𝑎�𝑜𝑓�𝐴𝑖 �≤ �𝑀𝑖 △ 𝑥𝑖 
This inequality applies to every 𝑖� = �1.2�,. . . ,𝑛�. So, from the sum of the minimum 
values obtained 

𝑚1 △ 𝑥1 + 𝑚2 △ 𝑥2 +.. . +�𝑚𝑛 △ 𝑥𝑛�𝑢𝑎𝑠𝑇ℎ𝑒�𝑎𝑟𝑒𝑎�𝑜𝑓�𝐴𝑖 
and from the sum of the maximum values obtained� 

𝐴𝑟𝑒𝑎�𝑜𝑓�𝐴𝑖 �≤�𝑀1 △ 𝑥1 + 𝑀2 △ 𝑥2 +. . .+�𝑀𝑛 △ 𝑥𝑛� 
The sum of the minimum values is referred to as the Lowersum and the sum of the 
maximum values is called the Uppersum. Analytically, it can be proven that if 𝑓�𝑖𝑠 
continuous at intervals [𝑎, 𝑏], then there is one number that satisfies the equation. 
Using GeoGebra, it can be proven that with more partitions being formed, i.e. with 
𝑛� → ∞, we will get exactly the same value from the uppersum and the lowersum. This 
same value is exactly the area of 𝐴𝑖. Visualization of the lowersum ad uppersum can 
be seen in the following figure: 

 
Figure 6 Visualization of Uppersum and Lowersum of a Function 

 

c. Definite Integral of Continuous Function 
Based on the area of the curve and the area of the curve and the distance calculation, 

then the problem is then abstracted into a continuous function. 

It is assumed that a function 𝑓�(𝑥) is defined and continuous at a finite closed interval 
[a, b]. Furthermore, given 𝑃 is a finite set of points arranged sequentially between 
𝑎�and 𝑏, i.e  
𝑃� = {𝑥0,𝑥1,𝑥2,𝑥3, . . . ,𝑥{𝑛−1} , 𝑥𝑛} 

with 𝑎� = 𝑥0 �<�𝑥1 < 𝑥2 < 𝑥3 <.. .< 𝑥{𝑛−1} < 𝑥𝑛 = �𝑏 
The set 𝑃�𝑖𝑠 called the partition of the interval [a, b] which divides [a, b] into n 
subintervals, where the i-th subinterval is [𝑥{𝑖−1} ,𝑥𝑖]. This subinterval is called the 
subinterval of the P partition, which can then be formed into rectangular arrangements, 

with the length and width being the length of the subinterval on the axis 𝑋 and its 
functional value on the axis 𝑌 as shown in the following figure: 



 

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Figure 7 Visualization of  𝑃 Partition on Interval [𝑎, 𝑏] 

 

 The length of the i-th subinterval of P is ∆�𝑥𝑖 =�𝑥𝑖 �−�𝑥{𝑖−1}, for 1≤i≤ 𝑛. 

Furthermore, for each subinterval [𝑥𝑖,𝑥{𝑖−1}], the function 𝑓 reaches the maximum 

value at 𝑀𝑖 and the minimum value 𝑚𝑖. 
Definition  

(Adams, 2010) Lowersum 𝐿�(𝑓,𝑃) and Uppersum 𝑈�(𝑓,𝑃) on  partition 𝑃 to the 
function 𝑓, defined as: 

𝐿(𝑓,𝑃) = 𝑚1𝛥�𝑥1�+�𝑚2𝛥𝑥2�+�.. .+𝑚𝑛∆𝑥𝑛 = ∑ 𝑚𝑖∆𝑥𝑖

𝑛

𝑖�=�1

 

𝑈(𝑓, 𝑃) = 𝑀1∆�𝑥1�+�𝑀2∆𝑥2�+�...+𝑀𝑛∆𝑥𝑛 = ∑ 𝑀𝑖∆𝑥𝑖

𝑛

𝑖�=�1

 

By Using Geogebra, we can calculate the lowersum and upersum of a function. We 

give the steps to calculate the lowersum and uppersum of a function by the following 

example  

Will be calculated  uppersum and lowersum value for the function f�(x)�=
1

x
  in the 

interval P� =�[1,2], which is divided into four subintervals with the same distance to 

know the value of its uppersum and lowersum. First defined the function 𝑓 at the input 
Bar  
>> f (x) = 1 / x 

Next, draw the boundary line interval, namely: 

>> x = 1 

>> x = 2 

If we want to find the partition in the interval [ 1,3], then register the partition points 

with the sequence command. The sequence command is useful for displaying values 

and visualizing those values on the cartesian axis. If partition points are displayed on 

the X-axis, the following command is used: 
>> Sequence [<Expression>, <Variable>, <Start Value>, <End Value>, <Increment>] become  

>> Sequence [(a + (i (b - a)) / n, 0), i, 0, n] 

In this case, 𝑎� +
𝑖(𝑏−𝑎)

𝑛
 is the 𝑛𝑡ℎ partition point formula, at intervals [𝑎,𝑏] with the 

same point distance. Then in the Algebra window, all partition points will be displayed 

in the form of coordinates on the X-axis in the Graphics window. And its values in the 

Algebra window.  

Furthermore, if we don't want to know the interval first, then the lowersum and 

uppersum values can be directly calculated, namely: 



 

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>> LowerSum [<Function>, <Start x-Value>, <End x-Value>, <Number of Rectangles>] 

>> UpperSum [<Function>, <Start x-Value>, <End x-Value>, <Number of Rectangles>] 

becomes  

>> LowerSum [f, 1,2,4] for the lowersum and 

>> LowerSum [f , 1,2,4] for uppersum  

Its calculation results can be seen in the window Algebra, and its figure on the 

Cartesian axes can be seen in the window Graphics 

The results can be visualized as follows: 

 
Figure 8 Calculation of Uppersum and Lowersum by using Geoebra 

 

According to the calculation of GeoGebra, then obtained points in the interval [1,2] 

can be seen on the Algebra display in the List partition, namely:  

𝑥0 = �1, 𝑥1 = �1.25, 𝑥2 = �1.25,�𝑥3 = �1.25,�𝑥4 = �1.25, 
Obtained Lower Sums (Lowersum) = 0.63 and Upper Sums (Uppersum) = 0.76. 
Meanwhile, using the definitio n of the lower and upper Sum, 
𝐿(𝑓,𝑃) = 𝑚1𝛥�𝑥1�+�𝑚2𝛥𝑥2�+�𝑚3𝛥𝑥3 + 𝑚4𝛥𝑥4� and  
𝑈(𝑓,𝑃) = �𝑀∆�𝑥1�+�𝑀2∆𝑥2�+�𝑀3∆𝑥3 + 𝑀4∆𝑥4,  
But the function value is calculated first at each point. To make it easier to know the 
value of its function, then you can add the following command GeoGebra: 
>> Sequence [<Expression>, <Variable>, <Start Value>, <End Value>, <Increment>] become  
>> Sequence [(a + (i (b - a)) / n, f (a + (i ( b - a))), i, 0, n] 

Then the points and function values given for each part of the point will be displayed, 
in Algebra's view, in the List Value Function, namely: 
𝑓�(𝑥0)�= �1, 𝑓�(𝑥1)�= �0.8, 𝑓�(𝑥2)�= �0.67,�𝑓�(𝑥3)�= �0.57,�𝑓�(𝑥4)�= �0.5 
With the value of the function, you will be able to calculate the Lower Addition and 
Top Addition based on the definition. The results obtained will be the same as the 

Lowersum and Uppersum results in the Figure above display.  
Based on the above example, either by calculation Lowersum and Uppersum on 

GeoGebra or by using the definition, the difference between 𝐿(𝑓,𝑃) = and 
𝑈(𝑓0.63,𝑃) =0.76, still fairly large. However, when the partitions of points added in 
this case 𝑛 gets bigger, the difference between 𝐿�(𝑓,𝑃) and 𝑈�(𝑓,𝑃) will be smaller.  
If more points are added to the interval, the bottom sum will increase, and vice versa 
the top sum decreases. Besides, the more the partition points, the difference in value 

between 𝐿�(𝑓, 𝑃) and 𝑈�(𝑓,𝑃) on the partition 𝑃 of the function 𝑓�𝑖𝑠 getting smaller. 



 

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So the values of 𝐿�(𝑓,𝑃) and 𝑈�(𝑓, 𝑃) will approach the area under the curve at interval 
[𝑎,𝑏] that is known. Visualization can be seen in the figure below: 

 
Figure 8 Calculation of Uppersum and Lowersum with different number of  𝑛 

 

Therefore, it can be proven that if 𝑓 is continuous on [𝑎,b], then there is exactly one 

number 𝐼 that satisfies the inequality 𝐿(𝑓,𝑃) ≤ 𝐼 ≤ 𝑈(𝑓,𝑃), for each partition 𝑃�in 
[𝑎,𝑏] 
Analytically, inequality can be proven as: 
 

Lemma: If 𝑃�and 𝑄 are partitions on [𝑎,𝑏], then 𝐿�(𝑓,𝑃)�≤ 𝑈�(𝑓,𝑄)[6] 
 

Proof: Take any 𝑃 ∪ 𝑄 partitions on [𝑎,𝑏] containing points the partition points at 𝑃 
and 𝑄, so that they apply: 𝐿�(𝑓,𝑃)�≤ 𝐿�(𝑓,𝑃 ∪ 𝑄)�≤ 𝑈�(𝑓,𝑃 ∪ 𝑄)�≤ 𝑈�(𝑓,𝑄) 

It is clear that 𝐿�(𝑓,𝑃)�≤ 𝑈�(𝑓,𝑄) 
From the lemma, it can be said that 𝐿�(𝑓,𝑃)�𝑖𝑠 limited to the top and 𝑈�(𝑓,𝑃)�𝑖𝑠 limited 
to the bottom. Since 𝐿�(𝑓,𝑃) limited to the above, there are a number 𝐿 so that if the 
result of the sum taken any number under 𝐿�(𝑓,𝑃) will always be less than or equal to 
a number. 𝐿�the In this case 𝐿�𝑖𝑠 called infimum from the sum below 𝐿�(𝑓,𝑃), where 
for any partition 𝑃� in [a, b], 𝐿�satisfies the inequality  

𝐿�(𝑓,𝑃)�≤ 𝐿 ≤ 𝑈�(𝑓,𝑃) 
In the same way, because 𝑈�(𝑓, 𝑃)�𝑖𝑠 limited to the bottom, there is a U number so that 
if any number is added to the sum of U(𝑓, 𝑃)�𝑖𝑡 will always be greater than or equal 
to the number 𝑈.�In this case, 𝑈�𝑖𝑠 called as a supremum from the sum below 𝑈�(𝑓,𝑃), 
where for any partition 𝑃� in [a, b], 𝑈�satisfies the inequality  

𝐿�(𝑓,𝑃)�≤ 𝑈 ≤ 𝑈�(𝑓,𝑃) 
Furthermore, it would be proved that if 𝑓 is continuous on [𝑎,b], then there is exactly 
one number 𝐼 that satisfies the inequality  
𝐿(𝑓,𝑃) ≤ 𝐼 ≤ 𝑈(𝑓, ) P, for each partition 𝑃�in [𝑎,𝑏] 
Theorem: 

If 𝑓� is continuous on [𝑎,b], then for each partition 𝑃�in [𝑎,𝑏] there is exactly one 
number 𝐼 that satisfies the inequality 𝐿(𝑓,𝑃) ≤ 𝐼 ≤ 𝑈(𝑓,)  
Proof:  

to prove the inequality, first demonstrated the existence of 𝑡ℎ𝑒�𝑓𝑖𝑟𝑠𝑡 number.  
Based on (2) and (3), then for each partition 𝑃 applies  

𝐿�(𝑓,𝑃) ≤ �𝐿 ≤ 𝑈 ≤ 𝑈�(𝑓,𝑃) 



 

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So there must be a number 𝐼 that satisfies, namely 𝐿 and 𝑈 itself.   
Furthermore, to prove the singularity of the number 𝐼�, it will be proven that 𝐿� = �𝑈 
Taken any > �0 
Because 𝑓� is a continuous function, then 𝑓� continuous is uniform at [𝑎, 𝑏]�𝑤ℎ𝑖𝑐ℎ 
means, there is𝛿 > �0, so ∀𝑥,𝑦 ∈�[𝑎,𝑏], with |𝑥𝑦| < 𝛿, then |𝑓(𝑥)− 𝑓(𝑦)| <  
Taken the partition 𝑃� =�{𝑥0,𝑥1, . . . . ,𝑥𝑛} with a maximum of ∆𝑥𝑖 < 𝛿 
So, for this partition 𝑃 applies  

𝑈(𝑓,𝑃) − 𝐿(𝑓,𝑃) = ∑ 𝑀𝑖∆𝑥𝑖 − ∑ 𝑚𝑖∆𝑥𝑖

𝑛

𝑖=1

𝑛

𝑖=1

 

= ∑ 𝑓(𝑢𝑖)∆𝑥𝑖 − ∑ 𝑓(𝑙𝑖)∆𝑥𝑖

𝑛

𝑖=1

𝑛

𝑖=1

 

= ∑(𝑓(𝑢𝑖) − 𝑓(𝑙𝑖))�

𝑛

𝑖=1

∆𝑥𝑖 

< ∑
𝑏 − 𝑎

�

𝑛

𝑖=1

∆𝑥𝑖 = 𝑏 − 𝑎
∑ �

𝑛

𝑖=1

∆𝑥𝑖 = 𝑏 − 𝑎
�(𝑏 − 𝑎) =  

Retrieved 𝑈(𝑓,𝑃) − 𝐿(𝑓,𝑃) <  
And since 𝐿�(𝑓,𝑃)�≤ 𝐿 ≤ 𝑈 ≤ 𝑈�(𝑓,P), the obtained 0 ≤ 𝑈𝐿 ≤ �𝑈(𝑓,𝑃) − 𝐿(𝑓, 𝑃) <  
In other words, 0 ≤ 𝑈𝐿� <  
it applies to every > �0, the obtained 𝑈𝐿� = �0 ⟺ 𝑈� = �𝐿 
Retrieved 𝑈� =L, which means the number 𝑓𝑖𝑟𝑠𝑡 that satisfies the inequality 
𝐿(𝑓,𝑃) ≤ 𝐼 ≤ 𝑈(𝑓,𝑃) is single. From the evidence of the existence and one number, 
𝑓𝑖𝑟𝑠𝑡 of the evidence that there are a number 𝐼,� which meets 𝐿(𝑓,𝑃) ≤ 𝐼 ≤ 𝑈(𝑓,𝑃) 
In GeoGebra, we can show numbers 𝐼� that meet these inequalities, by moving the 
slider 𝑛. The greater the value of 𝑛, the uppersum value will be smaller and the 
lowersum value will be greater, so that at a𝑛� certain, the Uppersum and Lowersum 
value will be closer to the same value. The closer the same value is, the closer the area 
under the curve is. This area of the same value will then become an integral value of 

the function 𝑓 at intervals [𝑎,𝑏]. From the numbers that satisfy the inequality, then it 
is defined Integral. 

 
Definitions of Definite Integral:  

Known 𝑓�𝑖𝑠 continuous at intervals [𝑎,𝑏].single number 𝐼 that satisfies the inequality 
𝐿(𝑓,𝑃) ≤ 𝐼 ≤ 𝑈(𝑓, ) P, for each partition 𝑃 in [𝑎,𝑏] called Fefinite integral (integral) 
function 𝑓 from 𝑎�to 𝑏 and is denoted as  

∫ 𝑥
𝑏

𝑎

�𝑓(𝑥)𝑑𝑥 

(Adams, 2010) 

If it exists with a single I that satisfies the inequality, then the continuous function 𝑓�𝑖𝑠 

called the function integrated in [𝑎,𝑏]. The symbol ∫ is an integral symbol that 

represents the letter 𝑆 from the word sum or sum, which indicates that the Integral is 
a representation of the sum as has been explained in the visualization and construction 
of definitions of the integral certainly above. Numbers 𝑎 and 𝑏 are integral limits (𝑎 
lower limit and 𝑏 upper limit of the integral). The function 𝑓 integrated is called the 



 

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integrant, 𝑥 is the variable integrated, 𝑑𝑥 as a derivative of 𝑥� which replaces the 
position ∆𝑥 in the lower and upper sums. If the integrand is a function that has more 
than one variable, the symbol is 𝑑𝑥 used as a reference to the variable to be integrated. 

For the function of one variable ∫ 𝑓(𝑥)𝑑𝑥
𝑏

𝑎
�, the variable 𝑥 can be replaced by another 

variable, as well as with 𝑑𝑥, even though some also write integral symbols without 
using 𝑑𝑥.  
The examples of  definite intergal writing of a single variable function are: 

∫ 𝑓(𝑥)𝑑𝑥
𝑏

𝑎
,�∫ 𝑓(𝑡)𝑑𝑡

𝑏

𝑎
,∫ 𝑓(𝑡)𝑑𝑡

𝑏

𝑎
,∫ 𝑓(𝑧)𝑑𝑧

𝑏

𝑎
,atau∫ 𝑓

𝑏

𝑎
saja.  

 

Visualization of Definite Integral Properties 

Acording to  definition of Definite integral, then we get the properties of Definite integral 

which are generally used in calculating the Integral of a function. These integral properties 

are based on the sum of the top and bottom of a function. Because the steps are quite long 

and use a lot of calculations (especially for partitions and high boundaries), then originally, 

the calculation of the integral value is certainly based on the following properties that are 

clearly by the results of the sum calculation above and below. The following will be 

presented with a visualization of the definite integral properties of course, by using 

GeoGebra. The visualization presented combines the calculation of the integral value of 

course based on the sum of the top and bottom sum of the functions. In GeoGebra, the 

calculation of integral values can certainly be done by inputting the values in the following 

command: 

>> Integral[ <Function>, <Start x-Value>, <End x-Value> ] 

 

Property 1: If any constant k,then ∫ 𝑘�𝑑𝑥
𝑏

𝑎
= 𝑘(𝑏 − 𝑎)�[7] 

With  GeoGebra, the ilustration as the following figure: 
 

 
Figure 9 Visualization of Definte Integral Property 1 

 

Property 2 :  Let 𝑓�be integrable function,the definite integral over an interval of zero 
length is zero  

∫ 𝑓(𝑥)�𝑑𝑥
𝑎

𝑎

= 0 



 

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With  GeoGebra, the ilustration is 

 
Figure 9 Visualization of Definte Integral Property 2 

 

Property 3 Reversing the limits of integration changes the sign of integral 

∫ 𝑓(𝑥)�𝑑𝑥
𝑎

𝑏

= − ∫ 𝑓(𝑥)�𝑑𝑥
𝑏

𝑎

 

With  GeoGebra, the ilustration is 

 
Figure 10 Visualization of Definte Integral Characteristic 2.3 

 

Property 2.1 Let f and 𝑔 integrable  on [𝑎,𝑏] and k is any constant, then 𝑘𝑓 and 𝑓 + 𝑔 
integrable on [𝑎,𝑏] : 

2.4.1 ∫ 𝑘𝑓(𝑥)
𝑎

𝑏
�𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥)

𝑏

𝑎
�𝑑𝑥 

2.4.2 ∫ [𝑓(𝑥)+ 𝑔(𝑥)]
𝑎

𝑏
�𝑑𝑥 = ∫ 𝑓(𝑥)

𝑎

𝑏
�𝑑𝑥 + ∫ 𝑔(𝑥)

𝑎

𝑏
�𝑑𝑥 

 

with  GeoGebra, the ilustration is 



 

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Figure 11 Visualization of Definte Integral Characteristic 2.4.1 

 
 

 
Figure 12 Visualization of Definte Integral Characteristic 2.4.2 (𝑥) and 𝑔(𝑥) 

 

The picture shows the results of the integral of the function f (x) and g (x), then the 

following is given a visualization of the integral results of the function f (x) + g (x) 



 

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Figure 13 Visualization of Definite Integral Characteristic 2.4.2 f(𝑥) + 𝑔(𝑥) 

 

The properties that have been visualized with GeoGebra above are some of the traits that 
are often used in solving problems in determining Integral values of course. It is also from 

these properties that theorems relating to Integral will emerge. For evidence in analysis, it 
may be usual to teach students, but not necessarily students understand the analytical 
meaning. 

By making the visualization on GeoGebra, it is expected that students' understanding of 
the meaning and things underlying the Integral is certainly. In addition to a clear picture, 

GeoGebra is also able to display dynamic visualization, thus allowing users to see a variety 
of changes in values that occur, as well as animations that make students more 
understanding and interested in understanding Mathematics. Students not only memorize 

the Integral formula given and then use it, but students know how to construct the material 
and what lies behind the appearance of the formulas and properties used. 

 

CONCLUSION 

The concept of Integral can certainly be abstracted from the problem of the area under the 

curve and the calculation of the distance of an object. Understanding the concept of 
definite integral using visualization of mathematical software such as GeoGebra will be 

able to interpret the meaning of the concept of mathematical material specifically definite 
integral. With the visualization of GeoGebra, it will be obtained that the integral can 
certainly be constructed from the lowersum and uppersum sum values of a continuous 

function f at intervals [a, b]. Where the integral certainly is a single value I that satisfies 
the inequality L (f,P) ≤I≤U (f,P), where L (f,P) is the lowersum and U (f,P) is uppersum. 

From the integral value, then comes the properties and theorems in Calculus which are 
then used as formulas in solving integral problems 
 

REFERENCES 

M. R. Caligaris, M.Graciela; Schivo, M.Elena; Romiti, “Calculus & GeoGebra, an 
Interesting Partnership,” Procedia-Social Behav. Sci., vol. 174, no. Sakarya 

University, pp. 1183–1188, 2014. 

D. Serhan, “Students’ Understanding of the Definite Integral Concept,” Int. J. Res. Educ. 
Sci., vol. 1, no. 1, pp. 84–88, 2015. 



 

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ISSN : 2579-5724   
ISSN : 2579-5260 (Online) 

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Contributors of Geogebra, “Geogebra Manual The Official Manual of Geogebra,” 

wiki.geogebra. 2015. 

Allan White, “Geogebra A Freeware Program (Geogebra Handout),” Sidney, 2016. 

C. Adams, Robert A.; Essex, Calculus Single Variable 7th edition. USA: Pearson, 2010. 

W. R. Taylor, A.E.; Mann, Advanced Calculus. USA: John Wiley, 1983. 

G. . Salas, S.; Hille, E.; Etgen, Calculus One and Several Variables,10th edition, 10th ed. 

USA: John Wiley&Sons, 2007.