Here is a problem that I was wondering if anyone could help me out on.
An ellipse has parametric equations
,
for
where
.
Show that its perimeter is given by the integral
.
Hence show that, if the ellipse is close to circular, the perimeter is approximately equal to
I managed to show that the perimeter is given by the given integral but I cannot make headway with the approximation. I rearrange the integral to give this;
I thought maybe I could integrate the binomial for the terms under the root symbol (using the maclaurin expansion of
but then that would take me to an answer with higher powers of
so I don't really know where to go with it.
Any thoughts would be appreciated!