Approximating the circumference of given ellipse

Question

Say we got the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{24}=1$, and the goal is to find the circumference using line integrals. So I parametrized the curve by $x=5\cos(t)$, $y=\sqrt{24}\sin(t)$. By definition we have that $S=\int_{c}ds$, so I wrote that $$S=\int_{0}^{2\pi}\sqrt{25\sin^{2}t+24\cos^{2}t} dt$$ However, I have been given the hint that $\sqrt{a^{2}c+b^{2}\left ( 1-c \right )}\geq ac+b\left ( 1-c \right )$, $0\leq c\leq 1$. I am not sure how to implement that into my equation for length. Maybe another parametrization would do the trick here? Also, the absolute error should not exceed 0.002.

Answer 1

Hint: $\sqrt{25\sin^{2}t+24\cos^{2}t}=\sqrt{24+\sin^{2}t}=\sqrt2\sqrt{24\left(\frac12\right)+\sin^2t\left(1-\frac12\right)}$