Riemann SumsA 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

The Necessity for Approximation

In previous courses, you’ve learned how to find the areas of regular geometric shapes using various formulas…

bhA2

1

lwA

212

1bbhA

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.

The BIG Unanswered Question???

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

0.4 width has oneeach ,rectangles 5 into subdivided isit Since

.2 to0for ofgraph theisgraph particular This 2 xxxy

left at the startingfunction theof value thefind toneed weSo, 0.4every then and 0,namely rectangle,first theofendpoint

.rectangles theof all of heights theus give willThis .after that

Left Hand Rectangular Approximation Method

6.14.02.14.0

8.04.04.04.004.05

ff

fffLRAM

have we, Since 2xxf

56.24.044.14.064.04.016.04.004.05 LRAM 92.1

widthby the heights theof oneeach multiply can weThen,

.rectangles theofeach of area thefind to

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

0.4 width has oneeach ,rectangles 5 into subdivided isit Since

.2 to0for ofgraph theisgraph particular This 2 xxxy

right at the startingfunction theof value thefind toneed weSo, 0.4every then and 0.4,namely rectangle,first theofendpoint

.rectangles theof all of heights theus give willThis .after that

Right Hand Rectangular Approximation Method

widthby the heights theof oneeach multiply can weThen,.rectangles theofeach of area thefind to

0.24.06.14.0

2.14.08.04.04.04.05

ff

fffRRAM

have we, Since 2xxf

44.056.24.044.14.064.04.016.04.05 RRAM 52.3

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

0.4 width has oneeach ,rectangles 5 into subdivided isit Since

.2 to0for ofgraph theisgraph particular This 2 xxxy

at the startingfunction theof value thefind toneed weSo,0.4every then and 0.2,namely rectangle,first theofmidpoint

.rectangles theof all of heights theus give willThis .after that

Midpoint Rectangular Approximation Method

widthby the heights theof oneeach multiply can weThen,.rectangles theofeach of area thefind to

8.14.04.14.0

0.14.06.04.02.04.05

ff

fffMRAM

have we, Since 2xxf

24.34.096.14.014.036.04.004.04.05 MRAM 64.2

Approximations from Numeric Data

These methods for approximation are most useful when we don’t actually know a function, but where we have several numerical data points that have been collected. In this situation, the widths of our rectangles will be determined for us (and may not all be the same).

Time Speed

0 3

3 8

5 14

11 7

15 18

Approximations from Numeric Data

Time Speed

0 3

3 8

5 14

11 7

15 18

To find the distance traveled in this situation, we can choose to use either LRAM or RRAM (but not MRAM since we don’t know the values at the midpoints. However, unlike our previous examples, the widths of our rectangles will all be different.

Approximations from Numeric Data

476142833 LRAM

Time Speed

0 3

3 8

5 14

11 7

15 18

To find LRAM, we will use the function value at the left endpoint and the varying widths of rectangles.

137LRAM

Approximations from Numeric Data

4186721438 RRAM

Time Speed

0 3

3 8

5 14

11 7

15 18

To find RRAM, we will use the function value at the right endpoint and the varying widths of rectangles.

166RRAM

Resource Pages

http://www.math.ucla.edu/~ronmiech/Java_Applets/Riemann/

http://www.synergizedsolutions.com/simpsons/pictures.shtml