009B Sample Final 1, Problem 5

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The region bounded by the parabola    and the line    in the first quadrant is revolved about the  -axis to generate a solid.

(a) Sketch the region bounded by the given functions and find their points of intersection.

(b) Set up the integral for the volume of the solid.

(c) Find the volume of the solid by computing the integral.

Foundations:  
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 cylindrical shells is given by

          where    is the radius of the shells and    is the height of the shells.


Solution:

(a)

Step 1:  
First, we sketch the region bounded by the given functions.
Step 2:  
Setting the equations equal, we have  
Solving for    we get
       
So,    and  
If we plug these values into our functions, we get the intersection points
         and  
This intersection points can be seen in the graph shown in Step 1.

(b)

Step 1:  
We proceed using cylindrical shells. The radius of the shells is given by  
The height of the shells is given by  
Step 2:  
So, the volume of the solid is

       

(c)

Step 1:  
We need to integrate

       

Step 2:  
We have

       


Final Answer:  
   (a)      (See Step 1 for the graph)
   (b)    
   (c)    

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