I've been given a problem, which I'm unsure whether I'm answering it right, and wondered if someone can take a look and tell me if I'm on the right track and any guidance would be appreciated.
My Problem:
Set up an integral that computes the circumference of an ellipse, but don't try to solve it as it is proven that the integral can't be solved. Where R is my radius.
Solution:
I know my integral should be something along the lines of: $$4\int_{0}^{R}\sqrt{a^{2}cos^{2}\theta+b^{2}sin^{2}\theta}d\theta$$
Is this along the right lines or am I meant to be using the formula: $$C=4aE(e)$$ where $e=\sqrt{1-\frac{b^{2}}{a^{2}}}$.
Any help and guidance on this would be great.
Thank you all in advance.
The length of a path $\gamma$ is given by $\int_a^{b} |\gamma'(t)|dt$. Here $\gamma (t)=(a \sin \, t, b \cos \, t), 0\leq t \leq 2\pi$. Hence the length is $\int_0^{2 \pi} \sqrt {a^{2} \cos^{2}(t)+b^{2} \sin^{2}(t)} dt$.