In determining the perimetre of ellipse one encounters the elliptic integral $$\int_0^{\frac{\pi}{2}}\!\!\sqrt{1-\varepsilon^2\sin^2t}\;dt,$$ where the parametre $\varepsilon$ is the eccentricity of the ellipse ($0 \leqq \varepsilon < 1$ ). A good upper bound for the integral is obtained by utilising the Cauchy-Schwarz inequality $$\left|\int_a^bfg\right| \;\leqq\; \sqrt{\int_a^bf^2}\,\sqrt{\int_a^bg^2}$$ choosing in it $f(t) := 1$ and $g(t) := \sqrt{1-\varepsilon^2\sin^2t}$ . Then we get