Application of  Integration to Calculate the Surface Area of Revolution

 

In a Nut Shell:  Calculation of surface area of revolution,  As , is based on the
Pythagorean Theorem.  The calculation typically involves three steps as follows:


Step 1    Visualize a “small” (differential) element, ds,  tangent to the curve, C, at an

arbitrary location  (x,y) as shown in the figure below.  The length  ds   can be calculated

using the Pythagorean theorem.

 

                       ds2  =  dx2  +  dy2       Thus    ds  =  √( dx2  +  dy2 )

 

                 axis of revolution

 

 

Step 2    For  y  =  y(x)        Write   ds  using x  as the independent variable.

 

         ds   =    [√ 1  +  (dy/dx)2 ] dx

 

The element of surface area,  dAs  =  2 π r  ds,   where       r  =  d + y

 

Note:  The element of area is just the circumference (2 π r) times the length of  ds.

 

 

Step  3   Determine the limits of integration in order to find the total arc length.  i.e.

 

                a  ≤   x    ≤   b        as shown in the figure above

 

Perform the integration to find the total surface area of revolution.   i.e.

 

                     b

    As   =  2 π ∫ (d + y) [√ 1  +  (dy/dx)2 ] dx         Click here for an example.

                     a

 

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