Air Pollution and Riemann Sums

 

(Adapted from the Accumulation project developed by Duke University’s Connected Curriculum Project, see http://math.duke.edu/education/ccp)

 

This project may be done in a group of up to 4 people.  Each member of the group will receive the same grade.  The project is due Tuesday, January 15.

 

Part 1: Background: Airborne Particulates

 

Air pollution is a serious problem which has serious health consequences. For example, it was recently estimated that particulate air pollution kills three times as many people in Houston, Texas, as die by homicides.

 

Particulate matter (PM) is one of six "criteria" air pollutants regulated by the Environmental Protection Agency (EPA) under the Clean Air Act and various amendments. The EPA is required to review the scientific research every five years to determine if stricter air quality standards are justified to protect the environment or the public health. Before 1987, the standard for PM made no distinctions in particle sizes, even though larger particles are filtered out by the nose, whereas smaller ones (10 microns in diameter or less -- a micron is one-millionth of a meter) can penetrate deep into the lungs. This "respirable particulate matter" (abbreviated PM10) includes liquids, hydrocarbons, soots, dusts, acids from aerosols, and smoke particles, sometimes accompanied by toxic chemical compounds that attach themselves to the particles. The current 24-hour National Ambient Air Quality Standard (NAAQS) for PM10, established by EPA in 1987, is 150 micrograms per cubic meter, with an annual average not exceeding 50 micrograms per cubic meter.

 

More recent research shows that even smaller particulates, those of 2.5 micron diameter or less, are even more dangerous than those between 2.5 and 10 microns. In July of 1997 the EPA adopted a PM2.5 standard.  The 24-hour standard is 65 micrograms per cubic meter, with an annual average not exceeding 15 micrograms per cubic meter.  (The National Resources Defense Council (NRDC) and American Lung Association (ALA) have recommended an annual PM2.5 standard of 10 μg/m3.)  The new standard was immediately challenged by industry groups, led by the American Trucking Association, who said that the EPA acted unconstitutionally in issuing the standards, and had failed to consider the cost to industries.  In May of 1999 the U.S. Court of Appeals for the District of Columbia remanded the new standards, meaning that the EPA had to withdraw them.  The decision was upheld by the full Court of Appeals in October of 1999.  However, the EPA continued the court battle, and in February of 2001 the U.S. Supreme Court unanimously upheld the tougher standards.  The EPA is now re-imposing the standards.

 

Here are some sources for the preceding paragraphs and related information:

 

·        Particulates Rank as a Leading Cause of Death in Houston (http://www.neosoft.com/~ghasp/articles/nrdcpm.htm)

·        Danger in the Air: Toxic Air Pollution in the Houston-Galveston Corridor (http://www.neosoft.com/~ghasp/toxics_report/phtp.htm)

·        Texas Environmental Almanac: National Ambient Air Quality Standards (http://www.texascenter.org/almanac/Air/AIRCH6P3.HTML#NATIONAL)

·        EPA ozone and PM2.5 standards upheld by the U.S. Supreme Court (http://www.dieselnet.com/news/0102epa.html)

 

Part 2: Summations: Size Distribution of Particles

 

Suppose we let p denote diameters of particles, and we let N(p) be the function whose value at p is the number of particles of size less than or equal to p in a cubic centimeter (cc) of air (notice that N(p) is strictly increasing). N is called the size distribution function, and its derivative dN/dp is called the size density function. We have no way to measure N directly, but we can measure dN/dp with reasonable accuracy in the following way: by using filters for particles of sizes p and p + Δp (the second one first), we can measure the number ΔN of particles between these two sizes. If the two sizes are close together, then ΔN/Δp is approximately the value of dN/dp at the particular size p.

 

Diameter p (microns)

dN/dp

0.00875

15700000

0.0125

5780000

0.0175

2580000

0.0250

1150000

0.0350

601000

0.0500

287000

0.0700

139000

0.0900

89000

0.112

70200

0.137

40300

0.175

25700

0.250

9610

0.350

2150

0.440

933

0.550

266

0.660

108

0.770

51.7

0.880

28.0

1.05

13.6

1.27

5.82

1.48

2.88

1.82

1.25

2.22

0.48

2.75

0.217

3.30

0.118

4.12

0.0627

5.22

0.0303

 

Above is data on the atmosphere over Pasadena, California, in August and September of 1969. The table gives values of the size density function dN/dp (computed as explained above) for various diameters, all less than 10 microns.  Enter this data into lists L1 and L2 on your calculator (accessed by STAT / EDIT / Edit – this means pressing the STAT key, looking in the EDIT menu, and choosing the option Edit).

 

1.      Make a scatterplot of the data for dN/dP as a function of p. Describe the graph.

 

2.      It's very difficult to get a good picture of data that vary as these do -- over nine orders of magnitude. You can change your window to look at smaller intervals on the p (i.e. x) axis to get a better sense of how the data varies.  In particular, look at the intervals [0, 0.1], [0.1, 1] and [1, 6] (you will have to choose an appropriate range for the y-axis in each case).  Describe the graph in each interval.

 

3.      Another way to plot data with such a large range of values is with a semilog (or log) graph. This means that, instead of plotting dN/dp against p, you plot ln(dN/dp) against p.  To create a new list of the values of ln(dN/dp), go to list L3, highlight the L3 on the top row, and press ENTER.  Then, at the bottom of the screen, enter L3 = ln(L2) (L2 is 2nd 2).  You can use this technique to create many new lists from old ones, without having to compute the entries one at a time.

 

Now plot the log graph and describe it.  Note that if y = Ceax, then the log plot (the plot of ln(y) against x) is a line.  What are the slope and intercept of this line (take the natural log of both sides of the equation y = Ceax)?  What can you say now about how the size density function varies? 

 

4.      Zipf's Rank-Size Law, originally formulated as a principle in linguistics, asserts that things of widely varying sizes tend to be distributed like a decaying exponential function. (Reference: Herbert A. Simon, "The Sizes of Things," in Statistics: A Guide to the Unknown, edited by Judith Tanur, Wadsworth & Brooks/Cole, 1985.) Are sizes of particles in the atmosphere distributed according to Zipf's law? Explain.

 

5.      In each size interval [p1,p2] we can count the number of particles:  (where Δp = p2 – p1). This is another instance of the familiar principle rise = rate times run. How many particles are there of size between 0.00875 and 0.0125 microns in a cc of air? Of size between 0.07 and 0.09 microns?

 

6.      If we want to count all the particles (of the sizes for which we have data) in a cc of air, we can add up all the rises for all the size intervals. Doing this one at a time would be very tedious – it is much easier to use list operations.  First, copy L1 to a new list, say L­4.  Delete the first entry of L4 and the last entry of L1.  L4 now contains the right endpoint of each size interval, and L1 contains the left endpoint.  Let L5 = L4 – L1, so L5 contains the differences Δp.  You can now create a list of the rises ΔN by letting L6 = L2 x L5 (delete the last entry of L2, so the lists are the same size).  Finally, use the command sum(L6) (2nd STAT / MATH / sum) to add up the entries in list L6.  What is the total number of particles (of the sizes for which we have data)?

 

7.      How many of the particles counted in the preceding step have diameters less than 2.5 microns? [You can use the same procedure as in Problem 6, only delete more entries from the lists.] What percentage of the particles have diameters less than 2.5 microns (i.e., would be subject to the PM2.5 standard)?  It’s clear why there was such strong industry resistance to the new PM2.5 standard!

 

Part 3: Accumulation and Area

 

In the preceding part, you calculated sums of the form

N'(p1) Δp1 + N'(p2) Δp2 + ... + N'(pn) Δpn.

That is, each term of the sum was a rate times a run, where the rate function N' (or dN/dp) was approximated by a difference quotient, and the corresponding Δp was the difference between consecutive diameters p.

 

Now let's interpret that sum geometrically. If we exaggerate the shape of the graph of N' a little -- so we can actually see it -- and pretend there were only five terms in the sum instead of 26 or 22, then we see that we have summed the areas of the rectangles in the following picture. (In this picture we are also pretending that the distances between consecutive values of p are all the same -- which was not the case with our particulate data.) The graph of N' is the curve shown. The height of each rectangle is a value of N' at the left edge of the corresponding rectangle, so we call the sum of the rectangular areas a left-hand sum.

 

 

Our left-hand sum with five terms is a rather crude approximation of the area under the graph of N'. But if there were 26 terms -- or 260 -- we could have a very good approximation to an area with a curved boundary.

 

We're going to explore that idea now for the graph of an arbitrary function with positive values. We will replace the p-axis with an x-axis and the function N' with y = f(x). Also, we will start the numbering of the points of subdivision with 0 instead of 1. This is just a matter of convenience -- it makes the largest subscript match the number of subdivisions (instead of being one larger). Here is the same picture with those notational changes:

 

 

The sum of the rectangular areas is now

f(x0) Δx + f(x1) Δx + f(x2) Δx + f(x3) Δx + f(x4) Δx

where Δx is the common width of the intervals, xk+1 - xk, for each k.  Of course, rather than using a left-hand sum we could use a right-hand sum, where the height of each rectangle is a value of N' at the right edge of the corresponding rectangle.  Left and right-hand sums are also known as Riemann sums.

 

1.      For the picture above, draw the rectangles of the corresponding right-hand sum, and write the algebraic expression for the right-hand sum.

 

We turn our attention now to an area problem for which you can calculate an exact answer. We will explore the extent to which left-hand (and corresponding right-hand) sums approximate this known area. Here is the figure whose area we will study:

 

 

2.      The shaded area in the figure (both regions) is the area under the graph of a function, namely, y = (25 - x2)1/2. On the other hand, it is also the area of a triangle plus the area of a sector of a circle. Use the latter description to find an exact symbolic expression for the total area. (First find the value of b.) Evaluate the symbolic expression to find a decimal approximation as well.  Be sure to show your work!

 

3.      Use the program RS which is included at the end of this project to compute left and right-hand sum approximations (LHS and RHS) to the shaded area with the interval [0, b] subdivided into n = 25 subintervals. How close is each approximation answer to the exact area?

 

4.      Now change n to 250, and recalculate both LHS and RHS. How close are these answers to the exact area?

 

5.      Change n again to 2500, and recalculate (it will take your calculator a couple of minutes). How close now? How much do the approximations improve each time you multiply n by 10?

 

6.      Which type of approximation, LHS or RHS, overapproximates the area? Which underapproximates it?

 

7.      Based on your answers to the two preceding steps, devise a procedure that would give better approximations than either LHS or RHS. Use your new procedure to compute approximations for n = 25, 250 and 2500.  How close are these approximations?  How much do your approximations improve when you multiply n by 10? How much better is your new procedure?

 

Part 4: Area Investigations

 

1.      Use your NEW approximation scheme to approximate the area of the region between x = 0 and x = π that is bounded above by the curve y = sin x and below by the x-axis. What do you think the exact area is?


 

2.      Use your NEW approximation scheme to approximate the area of the region between x = 0 and x = 3 that is bounded above by the curve and below by the x-axis.


 

Part 5: Summary

 

1.      If you have tabulated data on a rate-of-change function, how can you estimate the total change over an interval?

 

2.      Under what circumstances does a left-hand sum overestimate an area under a curve? Under what circumstances does it underestimate the area?

 

3.      Under what circumstances does a right-hand sum overestimate an area under a curve? Under what circumstances does it underestimate the area?

 

4.      How can you improve on LHS and RHS as estimates of the area under a curve? Is this "improvement" always better than both LHS and RHS? Explain.

 

Riemann Sum Program for TI-83

 

Create a new program at PRGM / NEW / Create New.

 

PROGRAM:RS

:0->S                                       (-> is the STO-> key)

:(B-A)/N->D

:A->X

:For(I,1,N,1)                            (PRGM / CTL / For)

:S+Y1->S                                 (VARS / Y-VARS / Function / Y1)

:X+D->X

:End                                         (PRGM / CTL / End)

:Disp S*D                                (PRGM / I/O / Disp)

:S+Y1->S

:A->X

:S-Y­1->S

:Disp S*D

 

Before running the program, store values for A, B and N.  A and B are the lower and upper bounds of the interval, N is the number of subdivisions.  Enter the function which bounds the region as Y1 on the Y= screen.  These values will not be changed by the program, so can be reused.  Run the program by selecting PRGM / EXEC / RS.  The program will display first the left-hand sum and next the right-hand sum.