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Introduction to Riemann sums
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Indefinite Integral versus Definite Integral
The area under a curve(A number)
A family of functions
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Born: About 287 BC in Syracuse, Sicily.Died: 212 or 211 BC in Syracuse.
Archimedes
Generally regarded as the greatest mathematician and scientist of antiquity.
He is called the "father of integral calculus" and also the "father of mathematical physics".
http://www.cs.drexel.edu/~crorres/Archimedes/contents.html
http://www.philosophyprofessor.com/images/philosophers/archimedes.jpg
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"Give me a lever long enough and a fulcrum strong enough,
and singlehandedly I will move the world." Archimedes
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http://math.furman.edu/~dcs/java/circle.html
Method of Exhaustion
Does this idea of inscribing and circumscribing polygonsto close in on the area of a circle remind you of a theoremwe have studied before?
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Images from: Math 122 Calculus for Biology II, Fall Semester, 2004, Riemann Sums and Numerical Integration, SDSU & Joseph M. Mahaffy
Review:Ways to approximate the definite integral:
1. Counting Squares
2. Trapezoidal Rule
http://www.csun.edu/~hcmth018/Trapez.html
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http://www.ecs.fullerton.edu/~mathews/a2001/Animations/Quadrature/Quadrature.html
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T6 = 1.4053...
All of T6 error is outside the curve.M6 error is both inside and outside of the curve (with more area excluded than included)So M6 is a more precise estimate of the true area.
M6 is an underestimate.Because the function is decreasing more rapidly on the left side of the midpoint than on the right side of the midpoint, there is more area left out and less area included. (See figure.)
T6 is an overestimate for the actual area under the curve. (See figure)
ci = 1, π/2, 2, 3, 4, 6.
ci = 0, 1, 3, 4, 3π/2, 5.c2 c5
sample point =
sample point =