Riemann Sums

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Riemann Sums
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Riemann Sums are used to approximate the area
between a curve and the x-axis over an interval.
Riemann sums divide the areas into rectangles.
By adding the areas of the rectangles, one gets
an approximation for the area under the curve on
the given interval.
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Usually Riemann sums will use equally sized
partitions of the interval to make calculations
easier. By having bases of equal length, the
base can be factored out when calculating the sum.
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Heres an example of how a Riemann Sum works
y1/x
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y1/x
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y1/x
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y1/x
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y1/x
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Base of each rectangle
Height of each rectangle using the right endpoint.
Think about it for a minute.
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Next, compute the Riemann sums for 50 and 100
intervals, using the same interval, 3,5.
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Taking it to....
the Next Level!
Now let's let the number of intervals approach
infinity.
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WOW!!! THIS LOOKS NASTY!!!
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YOU NOW HAVE A FORMULA FOR THE AREA UNDER THE
CURVE ON ANY INTERVAL a, b FOR THE FUNCTION y
x3 x2.
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You now know how to use Riemann Sums! The
next step is to add a bit of Calculus to the mix.
Georg Friedrich Bernhard Riemann Born September
17, 1826 Died July 20, 1866