Problem Diagram

Top View of the Planet's Orbital Path

Dr. Smith, a scientist at the Jetway Propulsion Laboratory, has recently discovered a planet orbit around a distant solar-type star named 55 Peg. The planet's path is found similar to an ellipse as shown in the figure. For a report on the new planet, he needs to estimate the distance traveled by the planet during a complete orbit.

Determine the arc length of one complete orbit.

• The parametric equations for the ellipse are given as x = 4 + 4cos(t) and y = 5 + 5sin(t) for 0 ≤ t ≤ 2π where x and y are in terms of 10⁸ km.
• Use Simpson's Rule to evaluate the integral if necessary.