MATH 132 – Lab 2 – January 18, 2005

Arc Length, and Surface Area

You should probably work on this lab in pairs. Each of you, though, should turn in your own report. This lab is due on Friday, January 21.

Part I: Arc Length

The arc length of a curve for a differentiable function f(x) on the interval [a, b] is given by . These integrals are often easy to set up, but hard to evaluate because the antiderivative is usually extremely difficult or even impossible to find in closed form. But we can use a computer algebra system like Derive to get exact or approximate answers.

1. A rope is to be hung between two poles 60 feet apart. The rope assumes the shape of the catenary , . Find the length of the rope (this is problem 20 from the homework), and find the “sag” in the cable; that is, the difference between the y-values in the middle (x=0) and the poles (x=±30). Be sure to state the integral you use explicitly in your lab report.

2. A fleeing object leaves the origin and moves up the y-axis. At the same time, a pursuer leaves the point (1,0) and always moves toward the fleeing object. If the pursuer’s speed is twice that of the fleeing object, the equation of the pursuer’s path is . Plot a graph of this situation. Where does the pursuer catch up with the object? How far has the fleeing object traveled when it is caught? Show that the arc length of the pursuer’s path is twice as far. Why does that make sense?

3. The Gateway Arch in St. Louis, Missouri is modeled by , for . Plot this curve, and find the length of this curve.

Part II: Surface Area

The surface area of a solid of revolution is given by where in the case where the solid is the region under a positive function and the region is rotated around the x-axis. Hence this integral becomes .

For each surface described below, set up the integral for the area of the surface of revolution, and approximate the value of integral.

4. Set up an integral to find the surface area of a sphere by revolving the circle about the x-axis. Evaluate the integral to find a formula for the surface area of a sphere.

5. Suppose the curve , for , is revolved about the x-axis (where R is some large positive constant).

a.) Find the volume inside the surface.

b.) Find the surface area of the surface.

c.) Find the limit of the volume as , and the limit of the surface area as .

d.) This result is sometimes referred to as the paradox of Gabriel’s horn. From your answers to exercise 5c, how much paint would it take to fill the horn? Again from exercise 5c, how much paint would it take to paint the outside of the horn? Do you see the paradox?