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Definite Integrals

Definite Integrals. As taught by Nathan "Definite" Shagam and Ethan "Integral" Scholl.

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Definite Integrals

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  1. Definite Integrals As taught by Nathan "Definite" Shagam and Ethan "Integral" Scholl

  2. Throughout time mankind has been fascinated by with what’s over. Over the horizon, over the edge, over the rainbow, have always intrigued us. However, now in this introduction to Chapter 1.3 you will venture into the deep, previously unexplored world of under. UNDER THE CURVE!!!!!!!! To help you understand this magical world, two masters of the land under the curve have been called to help. Nathan “Definite” Shagam and Ethan “Integral” Scholl are here to guide you on your quest of knowledge and understanding. Now, let us begin…..

  3. The formal meaning of a Definite Integral: The definite integral of the function f from x=a to x=b gives a way to find the product of (b-a) and f(x), even if f(x) is not a constant. What be Integral Basically, in more general terms, the definite integral of a function, is the area under the curve of that function from one x-point to another.

  4. Way to Find Definite Integral The most basic way to find the area under the curve (aka Definite Integral) and the first that you will be learning is counting squares. If you look at the picture of the coordinate plane to the left, you will notice that the grid lines for squares. By finding the area of one square, counting up the total number of squares under the curve from one x-point to another, and then multiplying those two numbers together, you will find the definite integral.

  5. With some curves, such as f(x)=x which is shown below, the squares under the curve are easy to count. However, with other more complicated functions such as f(x)= the amount of squares under the curve is much harder to approximate. This shows that counting squares is not a very good way to estimate the Definite Integral.

  6. Examples Question: Find the area under the curve of f(x)=x from x=2 to x=4. • Answer: • Find area of one square: Each square is 2x2 so the area of one square is 4. • Find the number of squares under the curve in-between those two x-points: There are 1.5 squares under the curve from x=2 to x=4. • Multiply the number of squares by the area of each square: 4(1.5)= 6. That is the Definite Integral.

  7. Question: Find the definite integral of f(x)= from x=1 to x=2. • Answer: • Area of one square = 1 • Number of squares 4 squares • Definite Integral = (4)(1) = 4

  8. Negative Area The area under the curve can be negative though. If the curve goes below the x-axis then Δy will be negative so the area will be negative. The same concept applies if you are finding the definite integral from x=2 to x=1, or where Δx is negative, but the curve of the graph is above the x-axis. Finally, if both Δx andΔy are negative then the area is considered positive.

  9. Other Applications In a velocity or acceleration graph, the definite integral is very important in finding certain information. Below are an acceleration and velocity graph. Acceleration (m ) Velocity (ms-1) Time (s) Time (s)

  10. If you took the definite integral of the acceleration graph from t=0 to t=4. What would that area represent? Well, the area would be ms-2 times s which equals ms-1. This shows that the definite integral or area under the curve of the acceleration graph is the velocity of the object at that time. So, by counting squares, the velocity of the object at time t=4 is 8ms-1. Acceleration (m ) Time (s)

  11. The same concept applies with the definite integral of a velocity graph. The definite integral of the velocity graph would be ms-1 times s, which equals m. Velocity (ms-1) Time (s) This shows that the area under the curve of a velocity graph represents the displacement of the object at a time. It represents displacement, because if there is negative area, that is direction which is a vector quantity, like displacement.

  12. Bibliography Images from the following websites: http://roberthood.net/blog/wp-content/uploads/2008/11/tremors-poster.jpg http://upload.wikimedia.org/wikipedia/commons/4/42/Integral_example.png http://ddrive.cs.dal.ca:9999/images/content/gamma_curve.jpg http://www.algebrahelp.com/calculators/function/graphing/ (for all the graphs) http://www.msstate.edu/dept/abelc/math/integral_area.png http://etc.usf.edu/clipart/49300/49310/49310_graph_blank_md.gif Programs Used: Winplot.exe Calc in motion (Co. Audrey Weeks www.calculusinmotion.com amweeks@aol.com

  13. Continued... Book used: Foerster, Paul A. Calculus: Concepts and Applications. 2nd ed. Emeryville: Key Curriculum P, CA.

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