Parametric Curves

A natural example is the path of a moving particle where f is time. We give a formula for the length of such a curve.

JUSTIFICATION

Figure 6.3.4

The infinitesimal piece of the curve (Figure 6.3.4) from t to t + Δt is almost a straight line, so its length Δs is given by

(compared to Δt),

(compared to Δt).

By the Infinite Sum Theorem,

The general formula for the length of a parametric curve reduces to our first formula when the curve is given by a simple equation

x = g(y) or y = f(x).

If y = f(x), a ≤ x ≤ b, we take x = t and get

If x = g(y), a ≤ y ≤ b, we take y = t and get

Example 2: Length of the path of a ball

The following example shows what happens when a parametric curve does retrace its path.

Example 3