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# 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 $e^6.$

## $L \approx 4 a\left[\frac{\pi}{2}-\frac{\pi}{8} e^{2}-\frac{3 \pi}{128} e^{4}+\frac{5 \pi}{512} e^{6}\right]$

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