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Problem : Find the average value of f (x) = | sin(x)| from x = 0 to 2Π.

We compute


=| sin(x)| dx  
 =sin(x)dx  
 =(- cos(x)|0Π)  
 =  

We could also have observed that in a previous section we found the area under this function to be 4, which is by definition the product of 2Π and the average value.

Problem : Suppose Eleanor invests 100 dollars in an account that is compounded continuously with an annual yield of 5 percent, so that the number of dollars in the account after t years is given by A(t) = 100(1.05)t. What is the average amount of money in her account over the first 3 years?

We have


=100(1.05)tdt  
 =|03  
 =(1.053 - 1)  

or approximately 107.69 dollars.

Problem : What is the average y-coordinate of a point on the upper half of the unit circle centered at the origin? (You may use that dx = (x/2) + (1/2)sin-1(x).)

The upper half of the unit circle centered at the origin is the graph of the function f (x) = on the interval [- 1, 1]. The average value of f on this interval equals


dx    
  = (2)dx  
 = + sin-1(x)|01=  

which is approximately equal to 0.79. Notice that this result depends critically on the interpretation of the word "average". Here we are thinking of average of the y-coordinate as a function of x. If instead we considered the y-coordinate as a function of arc length around the circle, we would be asking for the average of sin(t) on the interval [0, Π], which is 2/Π.
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