009B Sample Final 1, Problem 3

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Consider the area bounded by the following two functions:


(a) Sketch the graphs and find their points of intersection.

(b) Find the area bounded by the two functions.

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 graph these two functions.
009B SF1 3.png
Step 2:  
Setting    we get  
Therefore, we have
In the interval    the solutions to this equation are
Plugging these values into our equations,
we get the intersection points    and  
You can see these intersection points on the graph shown in Step 1.


Step 1:  
The area bounded by the two functions is given by


Step 2:  
Lastly, we integrate to get


Final Answer:  
   (a)      (See Step 1 above for graph)

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