007B Sample Midterm 2, Problem 4 Detailed Solution

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Find the area of the region bounded by    and  

Background Information:  
1. You can find the intersection points of two functions, say  

       by setting    and solving for  

2. The area between two functions,    and    is given by  

       for    where    is the upper function and    is the lower function.

3. Integration by parts tells us that


Step 1:  
We start by finding the intersection points of the functions    and  
So, we consider the equation   
The only solution to this equation is   
Also, for    we have


Step 2:  
The area bounded by these functions is given by


Now, we need to use integration by parts.
Let    and  
Then,    and  
Therefore, we have

Final Answer:  

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