Given is the Line Integral:
$$\int_C = \sqrt{\frac{a^2y^2}{b^2} + \frac{b^2x^2}{a^2}}ds$$
the Path $C$ is along the border of the ellipse with:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$$
and moves against Clockwise.
My attampt:
$$\int_C f(x,y) ds = \int_{t_1}^{t_2} f(x(t),y(t)) ||\frac{dr(t)}{dt}|| dt$$
with $$r(t) = (a\cos(t),b\sin(t)) \implies r'(t) = (-a\sin(t), b\cos(t)) \ \ \ \ t\in [0,2\pi]$$
Plugging this in and simplify comes out to:
$$I = \int_{0}^{2\pi} (a^2\sin(t)^2+b^2\cos(t)^2) dt$$
Now i am kinda stuck. I did not use the restricting condition of the ellipse yet. I tried to solve this restriction for either $a$ or $b$ and the plug it in, but this did not really work. Thank you for answering in advance.
