Vectors, Part 4

Vectors in Two and Three Dimensions

Part 4: Arc Length in the Plane

We can use parametric representation of curves and some of our results for vectors to compute the length of a curve represented by a vector function s(t) in the plane. In mathematical jargon, the length of a curve is called "arc length."

We begin by computing the infinitesimal length ds for a curve. In the following diagram, we approximate a small portion of a curve s by a small chord of length .

By the Pythagorean Theorem,

The next step is a little subtle. Since x and y are functions of t, we know from the differential approximation concept in single-variable calculus that the change in x is approximated by dx = x'(t)dt, and the change in y is approximated by dy = y^'(t)dt, with both approximations getting better as . Thus, if we write ds for the limiting value of , we see that, in the limit,

Here we have factored out the common factor of (dt)² under the square root sign and then brought it out from under the square root as dt.

The first factor of the right-hand side of this identity is the length of the tangent vector s^'(t). Now, just as with area under a curve, the total length of a curve between the points (x(t_i),y(t_i)) and (x(t_f),y(t_f)) determined by the initial t-value t_i and the final t-value t_f is found by summing all these "pieces" using the integral:

Use this arc length formula to compute the length of the curve drawn by (cos(t), sin(t)) for t from 0 to . How does this compare with what you know about the circumference of a circle? Can you do this without a computer?
Compute the arc length of the curve s(t) = (t, t²) for t-values from -5 to 5. Can you do this without a computer?

Compute the length of the orbit of Pluto in the following steps.

Let r(t) = (a cos(t), b sin(t)) be your parameterization. Why does this parameterize an ellipse in the plane?
In this parameterization, take a > b in your calculations. With this convention, a is the length of the semi-major axis of the ellipse, and b is the length of the semi-minor axis. Go to NASA's web site, and find the length of the semi-major axis of Pluto's orbit.
Using either kilometers (km) or astronomical units (1 AU = 149.6 million km = semi-major axis of Earth's orbit), compute the length of the semi-minor axis from the relation b² = a² (1 - e²), where e is the eccentricity of Pluto's orbit, which you can also find at NASA.
With the length of the semi-minor axis, determine the parameterization of Pluto's orbit.
Using your computer algebra system, compute the length of this orbit.