Chapter 7
The Fundamental Theorem of Calculus
7.1 Averaging Continuous Functions:
The Definite Integral
7.1.1 Average Temperature
Figure 1 Continuous temperature recording at Carlisle, PA, January, 1993
The weather report in your morning newspaper may contain a statement such as, "The average temperature yesterday was 60°." Temperature at a particular weather station is a continuously changing quantity, and it is often recorded continuously on a thermograph, like the one shown at the left. At every time `t`, there is a definite temperature, namely, the current reading on the weather station's thermometer. What does it mean to average a continuous function?
This is an important question for your gas and electric utility companies and your local heating oil supplier. In fact, the average temperature is the basis for running calculations of energy use based on heating and cooling degree days. By constantly monitoring energy use — at least the part determined by the weather — the utilities can anticipate short-run demand, and the oil supplier will know when to show up to deliver oil.
Figure 2 shows the hourly temperature readings at Baltimore Washington International Airport on December 18, 2006. This was an unusually warm day in Baltimore. The high of `72°` broke the previous record high for the date by three degrees. The weather bureau listed the low for the day as `47°` and the average temperature as `60°`.
Activity 1
Assuming that you had access to the temperatures at any time during the day (say from a recording thermograph), describe a method for determining the average temperature at BWI Airport on December 18, 2006.
In fact, what your newspaper means by "average" is the average of the high and low temperatures for the day. In this case, this average of `72°` and `47°` is `59.5°`, which was rounded up to `60°`. Even without knowing anything about averaging continuous functions, you probably devised a process for computing a more meaningful average temperature.
