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
• B
2. The Lower Limit• A
3. 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=b
6. 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
(2
4
14
14
x
1012
1012
xx
Cxxx
15
15
Property:
Cn
uduu
nn
1
1
(n ≠-1)
dxx 2
12
12
x
Cx
3
3
•Properties of Calculus
dxx 25
dxx 25
Cx
125
12
Cx
3
5 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. 3
ln 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