application of Cauchy–Schwarz inequality

In determining the perimetre of ellipse one encounters the elliptic integral

$${\int }_0^{\frac{\pi }{2}}\sqrt{1-{\epsilon }^2{\sin }^{2}t}𝑑t,$$

where the parametre $\epsilon$ is the eccentricity of the ellipse ().  A good upper bound for the integral is obtained by utilising the http://planetmath.org/node/1628Cauchy–Schwarz inequality

$$\left|{\int }_a^{b}fg\right|\leqq \sqrt{{\int }_a^b{f}^2}\sqrt{{\int }_a^b{g}^2}$$

choosing in it  $f(t):=1$  and  $g(t):=\sqrt{1-{\epsilon }^2{\sin }^{2}t}$.  Then we get

$\mathrm{\hspace{0.33em}}\leqq \sqrt{{\displaystyle {\int }_0^{\frac{\pi }{2}}}{1}^2𝑑t}\sqrt{{\displaystyle {\int }_0^{\frac{\pi }{2}}}\left(1-{\epsilon }^2{\sin }^{2}t\right)𝑑t}$

$\mathrm{\hspace{0.33em}}=\sqrt{{\displaystyle \frac{\pi }{2}}}\sqrt{{\displaystyle {\int }_0^{\frac{\pi }{2}}}\left(1-{\epsilon }^2\cdot {\displaystyle \frac{1-\cos 2t}{2}}\right)𝑑t}$

$\mathrm{\hspace{0.33em}}={\displaystyle \frac{\pi }{2}}\sqrt{1-{\displaystyle \frac{{\epsilon }^2}{2}}}.$

Thus we have the estimation

$${\int }_0^{\frac{\pi }{2}}\sqrt{1-{\epsilon }^2{\sin }^{2}t}𝑑t\leqq \frac{\pi }{2}\sqrt{1-\frac{{\epsilon }^2}{2}}.$$

It is the better approximation for the perimetre of ellipse the smaller is its eccentricity, i.e. the closer the ellipse is to circle.  The accuracy is $O({\epsilon }^4)$