TITLE: How do I evaluate this limit?
QUESTION [5 upvotes]: I'm trying to evaluate this limit:
$$\lim_{r\to 10}\int_0^r\sqrt\frac{{400-3x^2}}{400-4x^2}\ dx$$
at $x = 9.99999999999$, the value is $12.1105532$, so I feel like it is approaching some value around this. Does that mean there is a limit ? How do I evaluate that limit?

REPLY [4 votes]: One can see the limit results in the improper integral
$$L=\int_0^{10}\sqrt{\frac{400-3x^2}{400-4x^2}}~\mathrm dx$$
Now consider the substitution $x=10t$ to get
$$L=10\int_0^1\sqrt{\frac{400-300t^2}{400-400t^2}}~\mathrm dt=10\int_0^1\frac{\sqrt{1-\frac34t^2}}{\sqrt{1-t^2}}~\mathrm dt=10E(\sqrt{3}/2)$$
Where $E(k)$ is a complete elliptic integral of the second kind.