Finding the arc length of ellipses can be difficult. They involve Elliptic Integrals, hence the name.
The ellipse
can be expressed parametrically:
The total arc length of the ellipse
is given by
Where
. 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.