Calculus/Surface area

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Surface area

Suppose we are given a function and we want to calculate the surface area of the function rotated around a given line. The calculation of surface area of revolution is related to the arc length calculation.

If the function is a straight line, other methods such as surface area formulae for cylinders and conical frusta can be used. However, if is not linear, an integration technique must be used.

Recall the formula for the lateral surface area of a conical frustum:

where is the average radius and is the slant height of the frustum.

For and , we divide into subintervals with equal width and endpoints . We map each point to a conical frustum of width Δx and lateral surface area .

We can estimate the surface area of revolution with the sum

As we divide into smaller and smaller pieces, the estimate gives a better value for the surface area.

Definition (Surface of Revolution)Edit

The surface area of revolution of the curve   about a line for   is defined to be

 

The Surface Area FormulaEdit

Suppose   is a continuous function on the interval   and   represents the distance from   to the axis of rotation. Then the lateral surface area of revolution about a line is given by

 

And in Leibniz notation

 

Proof:

   
 
 

As   and   , we know two things:

  1. the average radius of each conical frustum   approaches a single value
  1. the slant height of each conical frustum   equals an infitesmal segment of arc length

From the arc length formula discussed in the previous section, we know that

 

Therefore

   
 

Because of the definition of an integral   , we can simplify the sigma operation to an integral.

 

Or if   is in terms of   on the interval