007B Sample Midterm 3, Problem 4 Detailed Solution

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Find the volume of the solid obtained by rotating the region bounded by    and    about the  axis. Sketch the graph of the region and a typical disk element.

Background Information:  

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
Squaring both sides, we get  
The solutions to this equation in the interval    are


Now, the graph of the region is below.
Additionally, we are going to be using the washer/disk method.
Below, we show a typically disk element.
(Insert graph)
Step 2:  
The volume of the solid using the disk method is

Final Answer:  
        See Step 1 for graph.

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