Finding the arc length of ellipses can be difficult. They involve Elliptic Integrals, hence the name.
The ellipse
![](/cgi-bin/mimetex.cgi?\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1)
can be expressed parametrically:
The total arc length of the ellipse
![](/cgi-bin/mimetex.cgi?x=a\cdot cos(t), \;\ y=bsin(t))
is given by
Where
![](/cgi-bin/mimetex.cgi?e=\frac{\sqrt{a^{2}-b^{2}}}{a})
. This e means eccentricity, not to be confused with the exponential e.
But, with yours maybe we can solve for y and use the arc length formula.
Knowing that arc length is given by
We could also try the polar and parametric arc length formulas:
In parametric:
Polar:
There you go. There is enough info to solve it one way or another.