Arclength and Surface Area



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


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

        y = x2  from 2 to 3.  


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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