# Computing the area under a curve

 Page 1 / 2
Basic Numerical Integration with MATLAB

This chapter essentially deals with the problem of computing the area under a curve. First, we will employ a basic approach and form trapezoids under a curve. From these trapezoids, we can calculate the total area under a given curve. This method can be tedious and is prone to errors, so in the second half of the chapter, we will utilize a built-in MATLAB function to carry out numerical integration.

## A basic approach

There are various methods to calculating the area under a curve, for example, Rectangle Method , Trapezoidal Rule and Simpson's Rule . The following procedure is a simplified method.

Consider the curve below:

Each segment under the curve can be calculated as follows:

$\frac{1}{2}({y}_{0}+{y}_{1})\mathrm{\Delta x}+\frac{1}{2}({y}_{1}+{y}_{2})\mathrm{\Delta x}+\frac{1}{2}({y}_{2}+{y}_{3})\mathrm{\Delta x}$

Therefore, if we take the sum of the area of each trapezoid, given the limits, we calculate the total area under a curve. Consider the following example.

Given the following data, plot an x-y graph and determine the area under a curve between x=3 and x=30

Index x [m] y [N]
1 3 27.00
2 10 14.50
3 15 9.40
4 20 6.70
5 25 5.30
6 30 4.50

First, let us enter the data set. For x, issue the following command x=[3,10,15,20,25,30]; . And for y, y=[27,14.5,9.4,6.7,5.3,4.5]; . If yu type in [x',y'] , you will see the following tabulated result. Here we transpose row vectors with ' and displaying them as columns:

ans = 3.0000 27.000010.0000 14.5000 15.0000 9.400020.0000 6.7000 25.0000 5.300030.0000 4.5000

Compare the data set above with the given information in the question .

To plot the data type the following:

plot(x,y),title('Distance-Force Graph'),xlabel('Distance[m]'),ylabel('Force[N]'),grid

The following figure is generated:

To compute dx for consecutive x values, we will use the index for each x value, see the given data in the question .:

dx=[x(2)-x(1),x(3)-x(2),x(4)-x(3),x(5)-x(4),x(6)-x(5)];

dy is computed by the following command:

dy=[0.5*(y(2)+y(1)),0.5*(y(3)+y(2)),0.5*(y(4)+y(3)),0.5*(y(5)+y(4)),0.5*(y(6)+y(5))];

dx and dy can be displayed with the following command: [dx',dy'] . The result will look like this:

[dx',dy'] ans =7.0000 20.7500 5.0000 11.95005.0000 8.0500 5.0000 6.00005.0000 4.9000

Our results so far are shown below

x [m] y [N] dx [m] dy [N]
3 27.00
10 14.50 7.00 20.75
15 9.40 5.00 11.95
20 6.70 5.00 8.05
25 5.30 5.00 6.00
30 4.50 5.00 4.90

If we multiply dx by dy, we find da for each element under the curve. The differential area da=dx*dy, can be computed using the 'term by term multiplication' technique in MATLAB as follows:

da=dx.*dy da =145.2500 59.7500 40.2500 30.0000 24.5000

Each value above represents an element under the curve or the area of trapezoid. By taking the sum of array elements, we find the total area under the curve.

sum(da) ans =299.7500

The following illustrates all the steps and results of our MATLAB computation.

x [m] y [N] dx [m] dy [N] dA [Nm]
3 27.00
10 14.50 7.00 20.75 145.25
15 9.40 5.00 11.95 59.75
20 6.70 5.00 8.05 40.25
25 5.30 5.00 6.00 30.00
30 4.50 5.00 4.90 24.50
299.75

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, A brief introduction to engineering computation with matlab. OpenStax CNX. Nov 17, 2015 Download for free at http://legacy.cnx.org/content/col11371/1.11
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'A brief introduction to engineering computation with matlab' conversation and receive update notifications?

 By By By By By By