Arclength and Surface Area

I.  Quiz

II.  Homework

III.  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

d = sqrt[(Dx)² + (Dy)²]  = sqrt[(Dx)²  [1 + (Dy/Dx)²]]

= sqrt[1 + (Dy/Dx)²]D x

Adding all the pieces together, we get

S sqrt[1 + (Dy/Dx)²]Dx

Taking a limit as Dx goes to zero we arrive at

For our example we have

= 70.9 feet

III.  Surface Area of Revolution

The area of a frostrum is 2pr(length)

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

Exercise:

Find the surface area of revolution about the x axis of the curve y = x² from 2 to 3.