Continuous function in two variables Let's have a look to the function $F(a,b)=\int_0^{2\pi}{\sqrt{a^2\sin^2(t)+b^2\cos^2(t)}dt}$, then this function describes the length of the arc of an ellipse ($a,b>0$). Can we see when this function $F$ is continuous on $(0,\infty)\times(0,\infty)$? I would say: if we can estimate the integral we are done. Is this right?
Thanks

The function $F$ is continuous at all points of the domain $(0, \infty) \times (0, \infty)$. Indeed, it's differentiable at all those points, and hence automatically continuous. (The derivatives are determined by "differentiating through the integral"). I'm not sure what you last sentence means, so I won't try to address it. 
