Thursday, March 26, 2015

To find the area of this surface, we rotate the function  about the y-axis (not the x-axis) in the range and this way create a finite surface of revolution.



A way to approach this problem is to swap the roles of and , essentially looking at the page side-on, so that we can use the standard formulae that are usually written in terms of  (ie, that usually refer to the x-axis).


The formula for a surface of revolution A is given by (interchanging the roles of  and )



Since we are swapping the roles of and , we need the function written as  in terms of  as opposed to in terms of . So we have



To obtain the area required by integration, we are effectively adding together tiny rings (of circumference at a point  on the y-axis) where each ring takes up length on the y-axis. The distance from the circular edge to circular edge of each ring is 


This is the arc length of the function in a segment of length of the y-axis, which can be thought of as the hypotenuse of a tiny triangle with width and height .


These distances from edge to edge of the tiny rings are then multiplied by the circumference of the surface at that point,  , to give the surface area of each ring. The tiny sloped rings are added up to give the full sloped surface area of revolution.


We have for this function, , that 


and since the range (in ) over which to take the integral is , or equivalently   we have  and  .


Therefore, the area required, A, is given by


  




So that the surface area of rotation A is given by


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