Annual Accumulation of Rainfall in California

Associate Prof. Geoff Hagopian, College of the Desert, 10/20/00.

Calculus IA.

Deadline for submission: one week from today.

This activity requires use of the Internet and spreadsheet software.

Safety Considerations
: Don’t get too attached to the numbers and don’t mess up the government’s web pages.

Teacher Preparation
: As usual, work through the project before you present it.

Objectives and Learning Outcomes
: Students will learn how the rainfall distribution function over both time (days and years) and geography (California) can be analyzed using the basic principles of calculus.

: The purpose of this project is to bring the tools of data acquisition via the Internet and the tools of calculus to bear on a relevant problem (rainfall distribution over time and geography) that will bring learning to all three endeavors: learning basic calculus; using the Internet for data acquisition; and rainfall distribution over time and geography.


  1. Point your Internet browser to the web site for the Climate Prediction Center of the National Oceanic and Atmospheric Administration (NOAA): and, in two hundred words or so, what the Climate Prediction Center is what sort of information they provide through the Internet.


  1. This site leads (circuitously) to a site titled, Southwestern U. S. Accumulated Precipitation, Actual vs. Normal Daily Precipitation, Last 365 Days:
    Click on the map’s dot for San Diego/Lindberg, California. Describe the information you find here by answering the following questions:
    1. What is the time frame of the data and what is rate of sampling for these data?
    2. Explain why both curves for accumulated rainfall (normal and observed) are strictly increasing.
    3. What is normal rainfall in mm per day during the rainiest part of the year in San Diego?
    4. Approximate the month and day on which the greatest daily observation of rainfall occurred and the month and day for which this normally occurs.
    5. Find the shortest time interval that includes 68% of the total rainfall.
    6. Assume that distribution of normal rainfall over the course of a year is "normal" in the sense that it can be modeled by a normal probability distribution. Find parameters A, m and s so that the function

      models the expected amount of rain at this San Diego location on a given day, t, in this time frame.
    7. How does R(t) compare with the observed rainfall?
    8. How does the curve for accumulated observed rainfall compare the curve for accumulated normal rainfall?
    9. Our function can be integrated to model the normal accumulated rainfall curve:

      where .  Plot A(t) and compare it the normal accumulated rainfall curve.
    10. What will A’(t) represent? What will its graph look like?
    11. Write a Riemann sum that approximates A’(t).


  1. Repeat the above steps for another weather station in California, your choice.


  1. Next, go to the California Department of Water Resources Division of Planning and Local Assistance: and then follow this (it’s not obvious, so just jump to the hyperlink given here) to the site for the climate data form: Here you should click “Monthly Data for the Previous Twelve Months” and then below that check the “San Diego” box and then click “Get CIMIS Data” at the bottom of the page.
    1. Import all these data (at least the rainfall data for San Diego) into an Excel spreadsheet and format to make it pretty and sensible.
    2. Use a Riemann sum construct in the Excel spreadsheet to find the total rainfall for San Diego for the period of time and compare it with the quantity observed in the web sites for #2.

:  Go to to see the current status of my solutions to the above. 

Teacher Reflections: I like it so far! “Upsides" include the use of real graphic and tabular data, software and the Internet to do calculus. “Downsides" include the time it took to prepare. I haven’t tried it out but I am looking forward to doing so.

Extensions: There is a lot of unused data here. There’s plenty of elbowroom for further exploration.

Assessment Suggestions: I would assign points as follows 10pts for #1; 50pts for #2;30pts for #3; 20pts for #4.

Standards Addressed: According to the AMATYC Standards for Introductory College Mathematics “Mathematics programs must demonstrate connections both among topics within mathematics and between mathematics and other disciplines.” Clearly, there is a connection with meteorology here.
Uri Treisman (University of Texas, Austin) is quoted in the Standards: "Art and music students at all levels have the opportunity to be creative. Mathematics students should have that same opportunity.” I think this activity allows the student some leeway for creativity.
Also, Standard I-6: Using Technology is clearly addressed in this activity.