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# ADDITIONAL MATHEMATICS PROJECT WORK 2017

## SABAH ENGLISH VERSION

PART 1

1. Importance of integration
Integration or also known as Integral calculus is the important part of calculus as important
differential calculus. In the differential calculus we study the relationship between two quantities
lets say between Distance and Time, this relationship we usually use in sense of Rate of Change
between two variables.

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But in Integral Calculus we take the inverse process of the relationship between two quantities, is
known as Integration or Anti-Differentiation or Anti- Derivative. The most important application of

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integral calculus is to compute the area or volume. In ancient time the informal concepts was

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developed by Greek Mathematicians Archimedes (287 BC 212 BC) and Eudoxus (410 BC 347 BC),
they developed the approximate area of different geometric shapes and these basic methods were

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also developed by Chinese Mathematician Liu Hui around 3th century as regards to find area of

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circle. In 17th Century John Kepler further developed some important concepts regarding to

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astronomical investigations to find area of sector and area of an ellipse.

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Further the concept of integral calculus was formally developed by Isaac Newton and Gottfried
Leibniz; they developed basic concepts to find area and volume. In integral calculus we encounter

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the different concepts such as area of various geometric shapes, area under the curve by using

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definite integral, indefinite integral and various practical applications. We also encounter the most
important theorem of calculus called Fundamental Theorem of Calculus this theorem elaborate

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the concept that differentiation and integration are opposite operations

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2. Uses of integration in different field.

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Integration is being widely used in different field. In engineering field, integration is used to develop
better navigation system. Apart from that, engineers use integral calculus for building skyscrapers,
and bridges. The Petronas Towers in Kuala Lumpur experience high forces due to winds. Integration
was used to design the building for strength. The Sydney Opera House is a very unusual design based
on slices out of a ball. Many differential equations (one type of integration) were solved in the
design of this building. Integration also been used in robotic manufacturing to make sure the robotic
part will work in given command. Apart for manufacturing, electrical and computer engineers used
integration for design a computer system. In term of automotive engineering, engineers use
integrals to improve the safety of vehicles.
In biological field, integral calculus is been used to study the interaction of different species within
an ecosystem. Other than that, the concentration of drugs in a living organism can only be answered
with integral calculus. As for backbones, integration plays major roles to measure the length of the
backbones and skulls. In microbiology, integration is used for measuring growth rate of bacteria and
certain species of viruses.

Apart from science field, integration is widely used in economic. This is been used in economic
research in term of relation of income, market prediction and costing.

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3. What I have learnt
Integration, Chapter 3 on first glance may look very complicated, but it was one of the challenging

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and interesting topics. In this chapter, first I have learnt the relationship between differentiation
and integration. Then I have learnt about Area under a Curve and area in between the two curves.

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Answer is by Integration. Other than that, I have also learnt the skills of finding volume of Solid of
revolution

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PART 2

1.

0.4 m
1m

t. k
1.2 m

## Height of table : 1 meter

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Width of table (excluding curve) = 1.2 meter
Curve height = 1 meter
Curve width = 0.4 meter
Width of table (including curve) = 2 meter
Scale of drawing is 1 : 20 o r
2. Equation of the curve of the table
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Simple sketch of the parabolic shape graph based on the dimension in (1)

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y = -a(x b)(x c)
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b=0, c=1
y = -a(x 0)(x 1)

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To find the value of a, substitute the coordinate (0.5, 0.4) into the equation

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0.4 = -a (0.5 0)(0.5 1)
a = 1.6

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Thus, the equation of the curve is y = -1.6x2 + 1.6x

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a
3. Area of the table curve is area under the graph.
b
Area y dx
a
1
Area (1.6 x 2 1.6 x) dx
0
1
1.6 x 3 1.6 x 2
Area
3 2 0
1.6(1) 3 1.6(1) 2
Area 0
3 2
Area 0.27m 2

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There are 2 curve sides, thus 0.27 m2 x 2 = 0.54 m2

## Area of rectangle = 1.2 m x 1 m

Area of rectangle = 1.2m2
Surface area of the table is the sum of area of 2 curve sides and rectangle
Area = 1.2 m2 + 0.54 m2
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Area = 1.74 m2
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PART 3

1.

Let say my watermelon has an ellipsoidal shape with major axis 28 cm and minor axis 25 cm.

a) Historical method
Because the melon is symmetrical, we can work out the volume of one half of the melon, and

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The radii for the slices for one half of a watermelon are found from measurement to be

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0, 6.4, 8.7, 10.3, 11.3, 12.0, 12.4, 12.5

2 2 2 2 2 2 2
Vhalf = x (6.4 , 8.7 , 10.3 , 11.3 , 12.0 , 12.4 , 12.5 ) x 2
= x 804.44 x 2
r
The approximate volume for one half of the melon using slices 2 cm thick would be:

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= 5054.4

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Thus, the volume of water melon is approximately 5054.4 x 2 half = 10108.8 cm3

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b) Formula =
x2 y2
1
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r
a2 b2
a = half length of the major axis
b = half length of the minor axis

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x2 y2
1
a2 b2
b 2 x 2 a 2 y 2 a 2b 2
a 2 y 2 a 2b 2 b 2 x 2 a t
b2
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a 2 y 2 b 2 (a 2 x 2 )

y 2 (a 2 x 2 )
d
2

a
a = 14 , b = 12.5

y2
12.5 2
2
a
Substitute in equation

(14 2 x 2 )
14
y 156.25 0.797 x 2
2

14
Vhalf y 2 dx
0
14
Vhalf (156.25 0.797 x 2 )dx
0

Vhalf 156.25 x 0.266 x 3 14
0

Vhalf 1457.60
Vhalf 4579.17cm 3
Thus the total volume of 2 sides of watermelon is 4579.17 cm3 x 2 = 9158.35 cm3
The volume calculated by using integration method is lower than the historical method which
differ by 950.45 cm3. This is because, in historical method, the shape is considered as circle not
as ellipse shape where ellipse shape is slightly smaller than circle. This proves that the historical
method gives a higher volume compared to the actual volume calculated by using integration
method.

## 2. a) Assuming the density of watermelon juice is 1 g/cm3

Based on the calculation above, the weight of watermelon is approximately
1g/cm3 x 9158.35 = 9.2kg
1 kg = 500 ml
9.2 kg x 500ml = 4600ml

## b) Based on price survey at Econsave, the price of watermelon is RM 1.90/kg

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Thus for 500ml, the cost of watermelon juice will be RM 1.90
1 litre of watermelon juice cost RM 3.80

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Selling price with 60 % profit

100 60
100
RM 6.08
3.80

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c) Price in Year 2017 = RM 1.90
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110
Price in Year 2018 = 100
1.90

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RM 2.09

110
2.09
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Price in Year 2018 = 100
RM 2.30
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110
Price in Year 2018 = 100
2.30

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RM 2.53

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In year 2020, 1 litre of watermelon juice cost RM5.06
Selling price with 60 % profit

100 60
100 m
5.06

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RM 8.10

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FURTHER EXPLORATION

In mathematics, we can simulate the creation of a bowl by using a graph function and then revolving

The height of my bowl is 5 cm and diameter is 10cm. and thickness is 0.5cm Based on this
information, a equation for the outer and inner is generated
y = 0.2x2 -5 (outer)
y = 0.22x2 -4.5 (inner)

## Below are the design using excel

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6
5

k
4
3
2
1 o r
0

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c
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-1
-2
-3
j e
-4

r o
p
-5

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-6

## Volume generated for outer

5
(0.2 x 2 5) 2 dx t h
5
5
(0.04 x 4 2 x 2 25)dx a
5

0.04 x 5 2 x 3

25 x
d m 5

400
3

3

a d
5

418.88cm 3
Volume generated for inner
4.5
(0.22 x 2 4.5) 2 dx
4.5
5
(0.04 x 4 2 x 2 25)dx
5
5
0.04 x 5 2 x 3
25 x
5 3 5
97.6896
306.90cm 3
Total material needed is Outer Inner
418.88 cm3 306.90 cm3 = 111.98 cm3
REFLECTION

## "The problem is all inside your head," he said to me.

The answer is easy if you don't forget plus C.
There must be fifty ways to do an integral.

## He said it's really not hard for you to do.

Furthermore, I hope my meaning won't be lost or misconstrued,
So I'll repeat myself At the risk of confusing you.
There must be fifty ways to do an integral.

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Refrain:
Use the FTC, Lee.

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Plug in a tan, Stan.
Do it by parts, Art,
But get the thing done.
Just plug in a u, Sue.
Use a table, Mable. o r
To set yourself free?
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He said you should know this better than your name.
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If only there were something to make you understand.
I said, "I appreciate that!
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