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Applications of the Integral

Problems

Average Value of a Function

Length of a Graph

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)|0 Π )  
  =  

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)t dt  
  = |0 3  
  = (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)|0 1 =  

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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