Worksheet #15 - Average Value
Spring 2007
Objectives
•Learning a new application of the integral
•Determining the average value of a function over an interval
Background You have taken the average of numbers before. This did not require
calculus. For practice, take the average of the following numbers. Consider the
process that it takes to take the average of a finite number of values.
30,28,25,22,18,15,19,22,23,24
The process is to sum the values and divided by the cardinality of the set. In
this case, the average is 22.6 (226/10).
Question How would you compute the average speed of your car over a certain
length of time? The speed of your car is a function of time and is not discrete.
But you could write down the car’s speed every 10 minutes or every 30 seconds
and take the average of the data that you collected.
Average Value of a Function Let’s consider the more general case. Suppose
you had a function f(t) and wanted to know the average value of the function
between t=aand t=b. Let’s pick Ntimes to measure the function and take the
average of these values.
a=t0, t1, t2, . . . , tn−1, tn=b
y0=f(t0), y1=f(t1), . . . , yN=f(tN)
Average Value = 1
NXyi=1
NXf(ti)
∆t=b−a
N
1
N=∆t
b−a
Average Value = ∆t
b−aXf(ti) = 1
b−aXf(ti)∆t
≈1
b−a( Area under the Curve )
=1
b−aZf(t)dt
