the chalkboard of integrals

The Chalkboard Of IntegralsMichael WagnerMegan Harrison

• Area Under A Curve• Sum of an infinite number of rectangles

The limit of an increasingly large number of increasingly smaller quantities, related to the function that is being integrated•What Is An Integral?

• Integrals have six parts1. The Upper Limit

• B2. The Lower Limit

• A3. The Function

• f(x)4. F(x) is the integral of f(x)5. F(b) is the value of the

integral at the upper limit, x=b6. F(a) is the value of the integral

at the lower limit, x=a

•What does it look like

• Bonaventura Cavalieri (1598-1647)

• Small rectangles under a line which would get so small they would be lines themselves. There are an infinite number of lines under a curve

• Gottfried Wilhelm Leibniz (1646 - 1716)

•Who invented Integrals

Sir Isaac Newton (1642 - 1727) • A defined fundamental theorem

• An indefinite fundamental theorem

•Integrals allow us to determine where an object lies at rest after being firedHow far did the rocket travel before is hit the ground?

•Why do we need integrals•Integrals give us a tool to quantify the things around us

How big are the Wasatch Mountains?How much dirt has been removed from Kennecott?

•Integrals allow us to determine the value of an item before we use itWhat is the maximum profit for a product?

•Integrals allow us to find the volume of an objectWhat is the volume of a vase?

•Properties of Calculus

dxx

x )11( 24

14

14

x1012

1012

xx

Cxxx

15

15

Property:

Cnuduun

n

1

1

(n ≠-1)

dxx 2

12

12

x

Cx

3

3

•Properties of Calculus

dxx 25

dxx 25

Cx

125

12

Cx

35 3

•Properties of Calculus..

Remember that if you just use these simple properties any integral is easy

•Multiple Choice Examples

1. xdxcos

a. Cx sin b. Cx sin c. Cx sec d. Cx csc

Hint: Remember that the derivative of sin(x) is cos(x)

•Mulitple Choice Examples

1. dxx 45

a. Cx

4

5

b. Cx

5

c. Cx 5 d. Cx 320

Hint: (x^(n+1))/(n+1)

•Multiple Choice Examples

1. dxx )1( 2

a. Cxx

3

3

b. Cx

2

3

c. Cxx 3 d. Cx

•Multiple Choice Examples

1. dxx 21

a. 2ln x

b. 3ln 3x

c. 1x

1 x

•Multiple Choice Examples

1. udusin

a. cosu C b. cosu C c. secu C d. cscu C

•Ha Ha Laugh

cabin1 dcabin = Ccabinln

http://www.math.wpi.edu/IQP/BVCalcHist/calc2.html

•Helpful websites

http://www.teacherschoice.com.au/Maths_Library/Calculus/calculate_definite_integrals.htm http://science.jrank.org/pages/3618/Integral.html

http://cs.smu.ca/apics/calculus/welcome.php

The End

Now you know a little bit of Calculus