Arclength and Surface Area

Arclength

Example

Suppose that you want to design a ski lift and need to determine how much cable will be needed between two posts that are 66 feet apart.  The formula for the cable is

        y = 50 cosh(x/50)

To compute the length of the cable we cut the cable into many pieces and the length of each piece is approximately equal to

Multiplying by (Dx)²/(Dx)² gives

Taking a limit as Dx goes to zero and adding up all the pieces, we arrive at

For our example we have

        =  70.9 feet

Surface Area of Revolution

The area of a frostum is 

        A = 2pr(length)

If we revolve a curve around the x-axis, we have that the surface area of revolution is given by

Example:

Set up an integral that gives the surface area of revolution about the x axis of the curve 

        y = x²  from 2 to 3.  


Solution

We find

        (y')² = (2x)² = 4x²

Now use the area formula:

       
We will learn later how to work out this integral (with much struggle).  However a computer gives that

        A @ 208.09

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