009B Sample Final 3, Problem 4

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Find the volume of the solid obtained by rotating about the  -axis the region bounded by    and  

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 a region around the  -axis using disk method is given by

          where    is the radius of the disk.


Step 1:  
We start by finding the intersection points of the functions    and  
We need to solve
If we square both sides, we get
The solutions to this equation are    and  
Hence, we are interested in the region between    and  
Step 2:  
Using the disk method, the radius of each disk is given by
Therefore, the volume of the solid is

Final Answer:  

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