INTEGRATIONDO NOT HESITATE, JUST INTEGRATE !!!! AP CALCULUS AB- GROUP PROJECT BY: WIOLETA WOJTYNA and JENNA ABOULAFIA

INTRODUCTION•Integration is one of the most important topics in calculus. It can be used in problems such as: finding the area between two curves, finding the volume of a figure using the washer or the disk methods, solving Riemann sum problems and finding the total distance and the average value. Moreover, integration is also widely known as anti-differentiation. This is a process that is opposite of finding the derivative. Together with differentiation, this mathematical process is the one that is used most frequently than any other topic.

Integration Can Be Used In ....•

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The Origins of Integrals •The earliest evidence of integrations dates

back to Ancient Greece. When mathematical analysis first began to develop, mathematicians would use what is called the Exhaustion Method to find areas of figures and volumes of solids. The Exhaustion Method is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape

The Origins of Integration cont.•At the end of the seventeenth century, the

works of Ferat, Newton, and Leibaiz pushed the theory of integrals into its primary stage. In the eighteenth century, the works of Euler, the Beroulli's, and Lagrange further developed mathematical analysis of the theory. Finally, by the nineteenth century, the theory of integral calculus was fully developed and complete.

IMPORTANT PEOPLE•

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Leonhard Nicolaus II Joseph Euler Bernoulli Lagrange

Definition of Integration•Integration means finding the anti-

derivative of a certain problem or undoing the derivative

 •Literally it is a process in which a person

combines or makes one thing out of two or more things.

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REAL WORLD PROBLEM• One week before the real AP Calculus exam, the AP

Calclulus exibit tickts went on sale at noon(t=0) and were sold within 5 hours. The number of people waiting on line is shown by the table below from

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T (time) 0 1 3 4 5

X(t) (people)

60 78 88 63 40

REAL WORLD PROBLEMa) Using the table above determine the rate at which the number of people waiting for the tickets was changing at t=2? Write the correct units and show all work.b) Using the trapezoidal sum with three subintervals estimate the average number of people waiting in line during the first 4 hours. Make sure to show all work.c) Using the data in the table, evaluate. Using correct units, interpret the meaning of in the context of the problem.d) The rate at which people are entering the AP Calculus exhibit is shown by the function: How many people entered the exhibit at t= 3?

METHODS a) For part a of this problem we have to use the Mean Value Theorem. We have to pick two points where t=2 would be in between.

Then plug in the given values into the equation above and solve for whatever the question is asking.b) For this part we have to use the trapezoidal sum.

Plug in the correct numbers into the equation and solve for the correct number of people that are waiting in line

METHODS c) For part c, we have to find the derivative of the function X (t) dt. We can find the derivative by using First Fundamental Theorem of Calculus.

For the meaning, since we are looking for the derivative then we know that it has to be the rate at which people are buying tickets.d) For this part we are given a function and we have to look for the people inside at t=4. We have to use the integral from t=0 to t=4 with the function inside. Since usually calculator is allowed we can just plug it into the calculator and record the answer.

Step by Step Solutions•    

 

=

This means the change in the rate at which people are buying the tickets from t= 0 to t=5. 

COMMON INTEGRALS

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PROPERTIES OF INTEGRALS

REFERENCES•Collegeboard.org•Mszhao.com•Google.com• http://

en.wikipedia.org/wiki/Method_of_exhaustion• http://

www.encyclopediaofmath.org/index.php/Integral_calculus