Laboratory I.10
It All Adds Up
Goals
• The student will work with Riemann sums and evaluate them using Derive.
• The student will see applications of integrals as accumulations of changes.
• The student will review curve fitting skills.
Before the Lab
The “Before the Lab” and “Ready for Lab” sections are longer than usual for this lab; be sure to
start early.
(On the other hand, the “In the Lab” and “After the Lab” are correspondingly shorter.)
Part I: The Initial Scenario
The Environmental Protection Agency recently investigated a spill of radioactive iodine.
Measurements
showed the ambient radiation levels at the site to be four times the maximum acceptable limit of 0.6
millirems/hour (abbreviated mrems/hr). The EPA ordered an evacuation of the area. You will investigate
several aspects of this incident using integrals and other basic information about functions.
At the beginning of the investigation, the emission rate was measured to be 2.4 mrems/hr.
Over the first
four hours, the following data were recorded:
There are some questions on this scenario in the “Ready for Lab?” section. If you need any help to
answer those questions, please study the question and answers on the last page.
Part II: Riemann Sums and Their Application in this Lab.
In addition to knowing the emission rate of the radiation, the EPA is also concerned with the total
radiation.
You can use the table above to estimate the total radiation in any one hour period as follows.
From the table of values we can assume that the radiation rate is getting smaller with time. For the first
hour, you know the radiation rate was at most 2.4 mrems/hr and at least 2.39041 mrems/hr.
So you
know that the total radiation emitted in that first hour was between 2.4 and 2.39041 mrems.
Note that
the upperestimate of 2.4 mrems was based on the reading at the start of the first hour, or the left
endpoint of the time interval [0,1].
Correspondingly, the lowerestimate of 2.39041 mrems was based
on the reading at the end of the first hour, or the right endpoint of the time interval [0,1].
The accumulation of the radiation emitted in all the individual hours is the total of all the radiation over
a longer time interval.
We will estimate the total radiation over the interval from time 0 to time 4, ie,
over [0,4]
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MATH 2413
Lab 10
Page 2 of 9
For each onehour interval, there is a left estimate of the radiation emitted in that hour based on the
radiation emission rate at the lefthand endpoint of that interval and a right estimate of the radiation
emitted based on the righthand endpoint of that time interval.
The total of the left estimates over the
total interval [0,4] is called the left Riemann sum (or lefthand sum, or LRS).
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 Spring '10
 Moody
 Integrals, Riemann Sums, Riemann sum, Riemann, right Riemann sum

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