By Marry Soegino and Jacky Wu

JOURNEY TO CALCULUS: AREA AND VOLUME

INTRODUCTIONI am Marry Soegino and is currently a junior in High School for Environmental Studies. I enjoy online shopping and online games when I have nothing else to do. My dream is to become a chemical engineer because I want to strive in a major where math and science is involved. My best subject in school is math and science, and I am really bad in English. When I grow up I hope to become heavily involved in changing people’s lives and shape the future of society.

INTRODUCTION

Hello! My name is Jacky Wu. I’m currently a student in High School for Environmental Studies. I enjoying eating the most, especially desserts of all kinds. I also enjoy solving math problems and helping others out. My dream is to be a one of the best computer engineer in the world.

PURPOSE• What is our propose?

• The purpose of our project is to find the volume of a star. Why a star? Because our universe are mostly made up of stars. One of the biggest star known to humankind is the Sun, the main source that creates life on Earth. But as we know it, there might be more than one Sun in the universe. By knowing the volume of a star, we could one day find more about stars and learn how it affects its surroundings and even life on Earth.

WHAT IS SIRIUS?• Sirius is a name of the brightest star

that people on Earth can see.• Sirius A, one of the stars in the binary

system of Sirius, is the biggest and brightest one. It is 3 times the mass of our Sun and 10 times as bright. Sirius B is the 2nd star relative to Sirius A. it is indeed smaller than Sirius A, but it has 4 times the density has Earth and has the same mass as the Sun.

• Sirius System is located above our solar system. Therefore its polarized energy can indeed affect Earth, but scientist has yet to found out how.

HISTORY OF VOLUME AND AREA

Around 250 B.C, Archimedes wrote a book called “On Conoids and Spheroids”. His book demonstrated how to found to find a volume of a parabaloid, the solid when revolving the parabola around its axis. This has greatly impacted the world of Calculus as of today.

DEFINITION OF VOLUME AND AREA• Definition of Volume and Area:

• Washer Method: The method of finding the inside volume of a object, by subtracting the outer volume and the inner volume of the object. It is given as a formula:

Area: Area is the space underneath a function. It is defined as the following:

FRQ: VOLUME, AREA, AND ROTATION AROUND A PLANET• In the universe, it is known that the brightest star is called

Sirius (aka Dog Star) and imagine the following star’s shape is represented by the equation:

y= 9cos(3x) and y= 5sin(2x)+7Now, MAJA wants to find the volume of the star as the star rotates around the x-axis at the 1st quadrant . • a) Find the area of Sirius using the given equation. • b) Find the volume of the star when Sirius is revolved around

the x-axis.

FRQ: VOLUME, AREA AND ROTATION AROUND A PLANET.• Graph of y = 9cos(3x) and y = 5sin(2x) +7

• Green: y= 5sin(2x)+7• Red = 9cos(3x)• Intersection Points: x= 1.7698802, x = 2.5316704

FRQ: VOLUME, AREA, AND ROTATION AROUND A PLANETa) Find the area of Sirius using the given equation:

We can use the equation: A=

Therefore we can substitute and get

By using a calculator, we can find the answer and get:A= 3.212

FRQ: VOLUME, AREA, AND ROTATION AROUND A PLANET.• B) Find the volume of the star as it rotates around the planet

represented by x-axis.• We can find this by using the Washers method, which is

Then we substitute :

Then we can find the answer by using a calculator.V= -13.877Because we are imagining it, the Volume would be positive, there fore V= 13.877.

REFERENCES

"Sirius." Dictionary.com. Dictionary.com, n.d. Web. 19 May 2013.

http://39clues.wikia.com/wiki/Archimedes

http://apcentral.collegeboard.com/apc/members/features/2015.html

http://www.siriusrising.com/sirius.htm

http://www.siriusstargazers.com/sirius-z.jpg

Mszhao.com

Collegeboard.com

http://www.wyzant.com/help/math/calculus/integration/finding_the_area

http://en.wikipedia.org/wiki/Disc_integration