Page 50 - Calculus for the Clueless

Chapter 9—
Parameters, Polar Coordinates Arc Length, Surface of Revolution

If the cartesian coordinate system were superior to all others, this chapter would not be necessary. However, it
is not. Parameters are variables that are introduced to make life easier.

Example 1—

We need to make a table with three columns' t, x, and y. We will use only the x and y on the graph.

t x y

-2 -4 0

-1 -2 -3/4

0 0 -1

1 2 -3/4

2 4 0

3 6 5/4

4 8 3

We can eliminate t: x = 2t; t = x/2; y = (x/2) /4 - 1 = x /16 - 1. It is clear we did not really need the parame ter t
2
2
in this case. However, we did not want to take an example that was too difficult in order to show the use of a
parameter.
Example 2—

2
2
Another common example is x = cos t, y = sin t. Again, we don't need a parameter, since x + y = 1.
Example 3—

Take a circle tangent to the x-axis at the origin. Put a dot on that point. Roll the circle on the x-axis. If we trace
out the path of that dot, we will get a curve called a cycloid. Since this curve is derived in most books, we will
put the equations down and then draw the picture.