Difference between revisions of "009B Sample Final 1, Problem 5"
Jump to navigation
Jump to search
Line 31:  Line 31:  
First, we sketch the region bounded by the given functions.  First, we sketch the region bounded by the given functions.  
    
−  [[File:  +  [[File:009B_SF1_5.pngcenter300px]] 
}  }  
Latest revision as of 12:16, 23 May 2017
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) 