Chapter 11.4, Problem 77E

Elliptic integrals The period of an undamped pendulum is given by

$T=\sqrt[4]{\frac{l}{g}}{\displaystyle \underset{0}{\overset{\pi /2}{\int }}\frac{d\theta }{\sqrt{1-k^2{\sin }^2\theta }}}=\sqrt[4]{\frac{l}{g}}F(k)$,

where l is the length of the pendulum, g = 9.8 m/s² is the acceleration due to gravity, k = sin $\frac{{\theta }_0}{2}$, and θ₀ is the initial angular displacement of the pendulum (in radians). The integral in this formula F(k) is called an elliptic integral, and it cannot be evaluated analytically. Approximate F(0.1) by expanding the integrand in a Taylor (binomial) series and integrating term by term.