Solution Diagram

Recall that when a curve is given using the following parametric equations

     x = f(t) and y = g(t) for a ≤ t ≤ b,

the arc length of the curve is defined as

The ellipse describing the planet's orbit is

     x = 4 + 4cos(t) and y = 5 + 5sin(t)
     for 0 ≤ t ≤ 2π

The first step is to evaluate the derivatives of the integral as,

The arc length of the ellipse is given by:

Since there is no obvious technique to perform the closed form integration, a numerical method called the Simpson's Rule is used. According to the Simpson's Rule, an integration can be approximated as follows:

where n is even and Δt = (b - a)/n

In this case, let , n = 10, and Δt = (2π)/10 = π/5

The values of f(t) subject to different values of t are summarized in the table on the left:

The integration can then be evaluated to give the distance of a complete orbit as