Chapter 11.10, Problem 98E

To determine

To expand: the integrand as a binomial series and express the length of the ellipse $L=4a{\displaystyle {\int }_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{1-e^2{\sin }^2\theta }d\theta }$ as a series in power of the eccentricity up to term in $e^6$ .