009B Sample Final 2, Problem 2

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Find the area of the region between the two curves    and  

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.


Step 1:  
First, we need to find the intersection points of these two curves.
To do this, we set
Getting all the terms on one side of the equation, we get
Therefore, we get that these two curves intersect at  
Hence, the region we are interested in occurs between    and  
Step 2:  
Since the curves intersect also intersect at    this breaks our region up into two parts,
which correspond to the intervals    and  
Now, in each of the regions we need to determine which curve has the higher    value.
To figure this out, we use test points in each interval.
For    we have
For    we have
Hence, the area    of the region bounded by these two curves is given by
Step 3:  
Now, we integrate to get

Final Answer:  

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