Chp5ishC
Riemann Sums Method
NOTES
Produces an approximation for the definite integral
1.
NEW METHOD FOR DETERMINING A
DEFINITE INTEGRAL
(an approximate value)
Given a function over the closed interval [a, b]
that does not produce a region where geometry
area formulas can be used it is possible to approximate its definite integral by using a method
derived mostly by a mathematician named Riemann, and his method is known as Riemann's
Sum.
(Shocking name, hunh?!)
In the area under the curve draw rectangles.
Get the area of each rectangle.
Get the sums of
the areas.
How to draw the rectangles under the curve can vary.
Look at BOTH pictures,
notice HOW the rectangles fit under the curve changes; sometimes they can be too short or too
tall.
For this example five rectangles are drawn.
LeftEndpoint
RightEndpoint
First
left endpoint
starts at x=a
Last right endpoint
finishes at x=b
Here the rectangles are too short
Here the rectangles are too tall
The
LEFT endpoint
of each rectangle
is on f(x)
The
RIGHT endpoint
of each rectangle
is on f(x)
The area is a small approximation – LOWER SUM
The area is a large approximation – UPPER SUM
RIEMANN SUMS is a method used to calculate an approximation of the area under a
curve by adding the areas of the rectangles formed.
The more rectangles drawn the
better the approximation.
NOTE:
Which endpoint (right or left) is used does not strictly determine whether area
approximated will be small or large.
It depends on the concavity of the curve of the interval
used.
For example, for a section of graph that is concave down where f(x) is increasing the left
endpoint rectanglesmake alower sum approximation of the area.
But, for a section of graph
that is concave up where f(x) is decreasing the leftendpoint rectangles makeanuppersum
approximation of the area.
f(x)
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 Spring '10
 KNOPF
 Approximation, Riemann Sums, Δx, rectangles, Riemann sum, Total Estim

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