Riemann Sum Example Problem

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Riemann Sums and Area

The curve in red in the pictures below is the graph

of y = x2 + 1.

The goal is to approximate and then compute

exactly the area between the x-axis and the graph

of y = x2 + 1 with x between 0 and 2. The region

being described is shown at the right in red.

Initially I will use four approximating rectangles.

As demonstrated in class, I will do it three times. I

will first use a left approximation, then a right

approximation, and then a midpoint

approximation.

Left

In the picture above the width of each rectangle will

be 1/2 and the length (height) will be the value of the

function evaluated at the left end of the subinterval

corresponding to that rectangle. In this case the area

of the region will be approximated by

Right



function evaluated at the right end of the subinterval



The left approach would give a low estimate and the right approach would give a high estimate.

Riemann Sum Example Problem http://www2.seminolestate.edu/lvosbury/CalculusI_Folder/RiemannSum...

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A pretty good estimate could be computed by taking the average of the left and right approximations.

A good estimate of the area of the region would

appear to be the area approximation derived from

the midpoint approach taken on the right. At least

this "looks" pretty accurate with the "top" of each

approximating rectangle partially above and

partially below the red graph of the function f(x) =

x2 + 1.

Midpoint



function evaluated at the midpoint of the subinterval



Let's try using more approximating rectangles and focus first on the right approximation.


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Quicktime Animation: right Riemann sum

At this point we can certainly conclude that the area of the region cannot be more than the limit

computed above.

It also cannot be less than than a limit we could compute relating to a left approximation method. Let's

see what that would be.


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Quicktime Animation: left Riemann sum

Just for fun let's look further at the midpoint method.

Quicktime Animation: midpoint Riemann sum


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What if we do not have a summation formula that applies? For example, how could we approximate the

area under the graph of y = sin(x) and above the x-axis with x between 0 and pi.

We will soon discover that the exact area of the region is 2 square units. Using the summation feature of

a TI-89 and n = 1000 yields 1.99999835507. In the picture above on the right n = 10 and the area

approximation is 1.983523538 using a TI-89 or Maple. The approximation for n = 2000 is 1.99999958876


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(TI-89). The accuracy of these "right" approximations is increased by the fact that the function is

increasing over the first half of the interval and decreasing over the second half of the interval. Thus

some of the approximating rectangles are too large and some are too small and the errors tend to balance

themselves. Quicktime animation using the midpoint method.

One More Example--Again Using a TI to Approximate the Sum. The approximated area for n = 10 is

16.26332364.

Here is a Maple worksheet for computing the sum above.

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Lane Vosbury, Math Chair, Seminole Community College email: [email protected] This page was last updated on 10/08/08 Copyright 2002 webstats


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Riemann Sum Example Problem

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