# Calculating the distance a satellite traveled

## Homework Statement

Im trying to figure out how to calculate how much a satellite in an elliptical orbit travels around in a day. The satellite is in a 12 day orbit with earth focused at (3,0). The equation i have for the ellipse is 16x^2 +25y^2 = 400. Other relevant information i have is the area that the satellite covered total every day, along with a and b coordinates for where it is every day, however I am having trouble cacluating the distance it travels day by day.

## The Attempt at a Solution

My first intution was to use the ∫ of √(25*sin(t)^2 + 16*cos(t)^2) dt and then just change the limits of integration, but im not sure if that would be right and what would i set the limits too?

Applying the Pythagorean Identity to your integral (which looks correct to me) gives us $\int_{t_{1}}^{t_{2}} \sqrt{25 - 9 \cos^{2} t } dt$ , which now has the typical form of such an elliptic integral. Unfortunately, not much can be done with this in exact form. You will need to use a numerical approximation method to get arclength values for each day, starting, say, from perigee.