In Exercise 10.2.53 it was shown that the length of the ellipse $ x = a \sin \theta, y = b \cos \theta, $ where $ a > b > 0, $ is

$ L = 4a \int^{\pi/2}_0 \sqrt {1 - e^2 \sin^2 \theta} $ $ d\theta $

where $ e = \sqrt {a^2 - b^2}/a $ is the eccentricity of the ellipse.

Expand the integrand as a binomial series and use the result of Exercise 7.1.50 to express $ L $ as a series in powers of the eccentricity up to the term in $ e^6. $