009B Sample Midterm 1, Problem 2

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Find the average value of the function on the given interval.


Foundations:  
The average value of a function on an interval is given by
.

Solution:

Step 1:  
Using the formula given in Foundations, we have:
Step 2:  
Now, we use -substitution. Let . Then, and . Also, .
We need to change the bounds on the integral. We have and .
So, the integral becomes .
Step 3:  
We integrate to get
   
Step 4:  
We evaluate to get
    .
Final Answer:  
   

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