Chapter 6.4, Problem 34ES

Show that the total arc length of the ellipse $x=a\cos t,y=b\sin t,0\le t\le 2\pi$ for $a>b>0$ is given by

$4a{\displaystyle {\int }_0^{\pi /2}\sqrt{1-k^2{\cos }^{2}t}}dt$

where $k=\sqrt{a^2-b^2}/a$ .