Help me calculate this integral
$\displaystyle \int_0^{2\pi} \sqrt{a^2\sin^2 x+ b^2 \cos^2 x}dx$
Thanks so much.
There is no known method for solving this integral explicitly. Your best bet is using a numerical method. Consider...
The length of an arc defined by the vector equation $\displaystyle \vec{x}(t)$ from t=A to t=B is $\displaystyle L=\int_A^B\sqrt{\left(\frac{dx}{dt}\right)^2+\left (\frac{dy}{dt}\right)^2}dt$
So the integral you have posed is equivalent to asking the circumference of the ellipse defined by $\displaystyle \vec{x}(t)=(-a\cos t,b\sin t)$. Take the derivatives to see what I mean. As you probably know, the circumference of an ellipse cannot be given in closed form.
You can either use an infinite series or approximation, which can be found here: Ellipse - Wikipedia, the free encyclopedia