Question

How do can you derive the equation for a circle's circumference using integration?

Answer 1

Let assume that the circle is centered at the origin hence its
equation is

`x^2+y^2=r^2`

Using parametrization in two variables: we can write the same circle above as

`(x(t),y(t))`,with `x(t)=rcost`,`y=rsint`,`0≤t≤2π`

and thus the arclength is given by (integration)

`\int_0^{2\pi}\sqrt{x'(t)^2+y'(t)^2}dt=\int_0^{2\pi}\sqrt{r^2\sin^2t+r^2\cos^2t}dt=r\int_0^{2\pi}dt=2\pi r`