8.2 Surface Area of Revolution Question #4 (Medium)

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8.2 Challenging Integral Applications
Surface Area of Revolution
Question #4 (Medium): Surface Area From Rotating the Function About the Y-Axis
Strategy
When the function is rotated about the -axis, the radius extends horizontally. When the function is
expressed as  , then 

. But if the function is given as  , then


. The expression from arc length can be evaluated with 
 just as well
because  
 
. The function   does not have to be rearranged,
and the derivative can be taken as it is.
Sample Question
Find the exact surface area obtained by rotating the given function about the -axis.
  ,   
Solution
Since the rotation is about the -axis, 

. But since the function is given for
in terms of , without having to rearrange, take the derivate as it is for 
, then


. So 
 . Thus:  

.
Since it is in chain rule form, substitution can be used. Let   , then  , so  
.
When  , then   , and when  , then  . Then:




Therefore, the surface area obtained by rotating the function   about the -axis over the
range    is
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Document Summary

Question #4 (medium): surface area from rotating the function about the y-axis. When the function is rotated about the -axis, the radius extends horizontally. When the function is expressed as , then ( But if the function is given as , then. The expression from arc length can be evaluated with just as well because ( The function does not have to be rearranged, and the derivative can be taken as it is. Find the exact surface area obtained by rotating the given function about the -axis. Since the rotation is about the -axis, ( But since the function is given for in terms of , without having to rearrange, take the derivate as it is for. Since it is in chain rule form, substitution can be used.

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