# Riemann Sums - PowerPoint

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```					Riemann Sums
A Method For Approximating
the Areas of Irregular Regions
Topics of Discussion
   The Necessity for Approximation
   Left Hand Rectangular
Approximation Methods
   Right Hand Rectangular
Approximation Methods
   Midpoint Rectangular
Approximation Methods
   Approximations from Numeric Data
   Comparing the Methods
A  lw               The Necessity for
Approximation
In previous courses,
you’ve learned
how to find the
areas of regular
geometric shapes
using various
formulas…

A  hb1  b2 
1
1                                    2
A  bh
2
The Necessity for
Approximation

However, when we have shapes like
these, there are no nice, neat formulas
with which to calculate their area…
The Necessity for
Approximation
To address this issue, we use a large number of
small rectangles to approximate the area of
one of these irregular regions. We may
choose to use rectangles with different widths
or rectangles with the same width.
The Necessity for
Approximation
Things to Remember
 When we know a function, it is best
to approximate the area using
rectangles with the same width.
 When we only know certain points

of the function, we will let those
points dictate the widths of the
rectangles that we use.
We’ve talked about the widths of the
rectangles that we will use for
approximation, but how will we
decide what the heights of these
rectangles will be?
Determining the Heights
of our Rectangles
Basically, unless specified, we can use
any point on the function in the given
interval for the height of a rectangle.
However, we typically choose to use
one of 3 points:
The Left Endpoint
The Right Endpoint
The Midpoint
Left Hand Rectangular
Approximation Method
As its name indicates, in the Left
Hand Rectangular Approximation
Method (LRAM), we will use the
value of the function at the left
endpoint to determine the heights
of the rectangles.
Left Hand Rectangular
Approximation Method

This particular graph is the graph of y  x 2 for x  0 to x  2.

Since it is subdivided into 5 rectangles, each one has width 0.4

So, we need to find the value of the function starting at the left
endpoint of the first rectangle, namely 0, and then every 0.4
after that. This will give us the heights of all of the rectangles.
Left Hand Rectangular
Approximation Method

Then, wecan multiply each one of the heights by the width
to find the area of each of the rectangles.

LRAM5  0.4  f 0  0.4  f 0.4  0.4  f 0.8 
0.4  f 1.2  0.4  f 1.6

Since f x   x 2 , we have

LRAM 5  0.4  0  0.4  0.16  0.4  0.64  0.4 1.44  0.4  2.56    1.92
Right Hand Rectangular
Approximation Method
As its name indicates, in the Right
Hand Rectangular Approximation
Method (RRAM), we will use the
value of the function at the right
endpoint to determine the heights
of the rectangles.
Right Hand Rectangular
Approximation Method

This particular graph is the graph of y  x 2 for x  0 to x  2.

Since it is subdivided into 5 rectangles, each one has width 0.4

So, we need to find the value of the function starting at the right
endpoint of the first rectangle, namely 0.4, and then every 0.4
after that. This will give us the heights of all of the rectangles.
Right Hand Rectangular
Approximation Method

Then, wecan multiply each one of the heights by the width
to find the area of each of the rectangles.

RRAM5  0.4  f 0.4  0.4  f 0.8  0.4  f 1.2 
0.4  f 1.6  0.4  f 2.0

Since f x   x 2 , we have

RRAM 5  0.4  0.16  0.4  0.64  0.4 1.44  0.4  2.56  0.4  4    3.52
Midpoint Rectangular
Approximation Method
As its name indicates, in the Midpoint
Rectangular Approximation
Method (MRAM), we will use the
value of the function at the
midpoint to determine the heights
of the rectangles.
Midpoint Rectangular
Approximation Method

This particular graph is the graph of y  x 2 for x  0 to x  2.

Since it is subdivided into 5 rectangles, each one has width 0.4

So, we need to find the value of the function starting at the
midpoint of the first rectangle, namely 0.2, and then every 0.4
after that. This will give us the heights of all of the rectangles.
Midpoint Rectangular
Approximation Method

Then, wecan multiply each one of the heights by the width
to find the area of each of the rectangles.
MRAM5  0.4  f 0.2  0.4  f 0.6  0.4  f 1.0 
0.4  f 1.4  0.4  f 1.8

Since f x   x 2 , we have

MRAM 5  0.4  0.04  0.4  0.36  0.4 1  0.4 1.96  0.4  3.24    2.64
Approximations from
Numeric Data
Time   Speed      These methods for approximation
are most useful when we don’t
0      3
actually know a function, but
3      8            where we have several numerical
data points that have been
5      14           collected. In this situation, the
widths of our rectangles will be
11     7            determined for us (and may not
all be the same).
15     18
Approximations from
Numeric Data
To find the distance traveled in
Time   Speed
this situation, we can choose to
0      3          use either LRAM or RRAM (but
not MRAM since we don’t know
3      8          the values at the midpoints.
5      14         However, unlike our previous
examples, the widths of our
11     7          rectangles will all be different.
15     18
Approximations from
Numeric Data
To find LRAM, we will use the
function value at the left endpoint
and the varying widths of              Time   Speed
rectangles.                            0      3
LRAM  3  3  8  2 14  6  7  6   3      8
LRAM  141                             5      14

11     7

15     18
Approximations from
Numeric Data
Time   Speed       To find RRAM, we will use the
0      3
function value at the right
endpoint and the varying widths
3      8           of rectangles.
5      14

11     7          RRAM  8  3  14  2  7  6 18  4
15     18         RRAM  166
Comparing the Methods
For each of the functions on your
“Riemann Sum Worksheet”, use
the Riemann Sum Calculator
LRAM, RRAM and MRAM for 5,
10, 25, 50 and 100 rectangles.

For Riemann Sum Calculator
Resource Pages
   http://www.math.ucla.edu/~ronmie
ch/Java_Applets/Riemann/
   http://www.synergizedsolutions.co
m/simpsons/pictures.shtml

```
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 views: 2 posted: 2/28/2012 language: pages: 23