009B Sample Final 2, Problem 3

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Find the volume of the solid obtained by rotating the region bounded by the curves    and    about the line  

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

        by setting    and solving for  

2. The volume of a solid obtained by rotating an area around the  -axis using the washer method is given by

          where    is the inner radius of the washer and    is the outer radius of the washer.


Step 1:  
First, we need to find the intersection points of    and  
To do this, we need to solve
Moving all the terms on one side of the equation, we get
Hence, these two curves intersect at    and  
So, we are interested in the region between    and  
Step 2:  
We use the washer method to calculate this volume.
The outer radius is
and the inner radius is
Therefore, the volume of the solid is
Step 3:  
Now, we integrate to get

Final Answer:  

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