##### Document Text Contents

Page 2

Candidate: 000738-0003 Arandia Jimenez 2

multiplying the dimensions of a geometrical three-dimensional shape. Considering its purpose, I

wondered: Could be possible to calculate the volume of non-regular shapes, such as a vase?

Figure 1. shows a sine function on a graph with its shaded area under the line, and Figure 2.

shows how it looks when the area is rotated 36 around the x-axis.00

Figure 1. and 2; “Solid of Revolution - Finding Volume by Rotation”; WyzAnt; wyzant.com;

Web; 10 March, 2016.

r is changing for every point in the x-axis, and if r ¿ f ( x )= y , the area of the imaginary

cross sectional circle inside the vase will also be changing, because it is an irregular 3D shape. In

order to find the volume of the vase using the area ( A=π r

2

¿ of its imaginary cross sectional

disks inside, the volume of a cross sectional disk has to be calculated. The area of the disk should

be multiplied by its infinitesimally small thickness (dx).

V d=π r

2

dx

Next, we have to integrate in order to find the volume of the entire base. Integration will

add up the volume of all disks that form the vase, considering that the volume of each is

changing for all values of x. The area under the function f(x) will be rotated around the x-axis,

with limits established by vertical lines x=a and x=b. (“Clip 2: Solids of Revolution”).

V = ∫

b

a

π r

2

dx → V = ∫

b

a

π ( f (x))

2

dx

Figure 1. Sketch of a sine function on a set of axis Figure 2. Sketch of a rotated sine functions

on a set of axis on a set of axis

Page 27

Candidate: 000738-0003 Arandia Jimenez 27

V 3=V II ( x )−V j(x )

V 3=63.5392 cm

3

The three volumes found have to be added to find the volume of the entire flower vase:

V=V 1+V 2+V 3

V=1533.0094+199.7362+63.5392

V=1796.2848 cm

3

V. Conclusion and Reflection

Throughout the process of this exploration I was able to learn new mathematical concepts, as

well as apply previous knowledge on a real life situation. The new skill I learned was how to find

points on a graph that follow the Golden spiral and the Golden ratio. Not only have I practiced

finding the volume of 3D solid objects by integrals, but I also improved my skills on finding the

volume of irregular solid objects that require integrating more than 3 functions to be solved. For

me, this mathematical exploration was a proof that the disc method on a solid of revolution can

be applied in a real life situation or possible real life situation. This means that this method can

be used to find the volume of several, and different, irregular solid objects, such as bottles and

even more types of flower vases. There are so many objects that can be modeled to find its

volume that we, humans, will never get tired of applying calculus in several and unique forms!

My aim of finding the volume of a non-conventional object was fulfilled. After achieving my

aims, I was thinking where else I could apply the processes of modeling and integrating to find

volume of this flower vase; I wanted to find a real life situation where this would be useful. The

first idea that came to my mind was that I could use it to optimize the price of it. This means that

I could figure out what is the least or greatest amount of material (glass for example) that could

be used to create this “proposed” flower vase, and then find the cost of production by finding the

price of the material per cm

3

, and multiply it by the volume calculated. This process is done

Candidate: 000738-0003 Arandia Jimenez 2

multiplying the dimensions of a geometrical three-dimensional shape. Considering its purpose, I

wondered: Could be possible to calculate the volume of non-regular shapes, such as a vase?

Figure 1. shows a sine function on a graph with its shaded area under the line, and Figure 2.

shows how it looks when the area is rotated 36 around the x-axis.00

Figure 1. and 2; “Solid of Revolution - Finding Volume by Rotation”; WyzAnt; wyzant.com;

Web; 10 March, 2016.

r is changing for every point in the x-axis, and if r ¿ f ( x )= y , the area of the imaginary

cross sectional circle inside the vase will also be changing, because it is an irregular 3D shape. In

order to find the volume of the vase using the area ( A=π r

2

¿ of its imaginary cross sectional

disks inside, the volume of a cross sectional disk has to be calculated. The area of the disk should

be multiplied by its infinitesimally small thickness (dx).

V d=π r

2

dx

Next, we have to integrate in order to find the volume of the entire base. Integration will

add up the volume of all disks that form the vase, considering that the volume of each is

changing for all values of x. The area under the function f(x) will be rotated around the x-axis,

with limits established by vertical lines x=a and x=b. (“Clip 2: Solids of Revolution”).

V = ∫

b

a

π r

2

dx → V = ∫

b

a

π ( f (x))

2

dx

Figure 1. Sketch of a sine function on a set of axis Figure 2. Sketch of a rotated sine functions

on a set of axis on a set of axis

Page 27

Candidate: 000738-0003 Arandia Jimenez 27

V 3=V II ( x )−V j(x )

V 3=63.5392 cm

3

The three volumes found have to be added to find the volume of the entire flower vase:

V=V 1+V 2+V 3

V=1533.0094+199.7362+63.5392

V=1796.2848 cm

3

V. Conclusion and Reflection

Throughout the process of this exploration I was able to learn new mathematical concepts, as

well as apply previous knowledge on a real life situation. The new skill I learned was how to find

points on a graph that follow the Golden spiral and the Golden ratio. Not only have I practiced

finding the volume of 3D solid objects by integrals, but I also improved my skills on finding the

volume of irregular solid objects that require integrating more than 3 functions to be solved. For

me, this mathematical exploration was a proof that the disc method on a solid of revolution can

be applied in a real life situation or possible real life situation. This means that this method can

be used to find the volume of several, and different, irregular solid objects, such as bottles and

even more types of flower vases. There are so many objects that can be modeled to find its

volume that we, humans, will never get tired of applying calculus in several and unique forms!

My aim of finding the volume of a non-conventional object was fulfilled. After achieving my

aims, I was thinking where else I could apply the processes of modeling and integrating to find

volume of this flower vase; I wanted to find a real life situation where this would be useful. The

first idea that came to my mind was that I could use it to optimize the price of it. This means that

I could figure out what is the least or greatest amount of material (glass for example) that could

be used to create this “proposed” flower vase, and then find the cost of production by finding the

price of the material per cm

3

, and multiply it by the volume calculated. This process is done