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

Solved: "How do can you derive the equation for..."

racodelitusmn

2022-07-14

Shawn Castaneda

Beginner2022-07-15Added 17 answers

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) = r \cos t, y = r \sin t, 0 \leq t \leq 2 π

and thus the arclength is given by (integration)

\int_{0}^{2 π} \sqrt{x^{'} (t)^{2} + y^{'} (t)^{2}} d t = \int_{0}^{2 π} \sqrt{r^{2} \sin^{2} t + r^{2} \cos^{2} t d t} = r \int_{0}^{2 π} d t = 2 π r

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