TITLE: Integration for finding the Arc Length of Circle $x^2+y^2=a^2$
QUESTION [6 upvotes]: Question:

Find the arc length of the circle given by $x^2+y^2=a^2$.   
$Ans = 2\pi a$

How to obtain the ans? I have no ideas after doing the following thing.

Thank you for your attention.

REPLY [2 votes]: Consider the figure below (apologies for the poor formatting)

We can use the following:-
$$L=\int_0^{2\pi}\sqrt{\left(\frac{dX}{d\theta}\right)^2+\left(\frac{dY}{d\theta}\right)^2}d\theta$$
where $X=a\cos\theta$ and $Y=a\sin\theta$
So we have
$$\frac{dX}{d\theta}=-a\sin\theta,\frac{dY}{d\theta}=a\cos\theta$$
resulting in
$$L=\int_0^{2\pi}\sqrt{a^2\cos^2\theta+a^2\sin^2\theta}d\theta\\=\int_0^{2\pi}\sqrt{a^2(\cos^2\theta+\sin^2\theta)}d\theta\\=\int_0^{2\pi}(a)d\theta=\left[a\theta\right]_0^{2\pi}=2\pi a$$
where we have used the identity that $\sin^2\theta+\cos^2\theta=1$