3.5 Length of a curve

We think of $ds^{2}=dx^{2}+dy^{2}$ and write $L=\int ds.$ Let $P$ and $Q$ be points on a curve $C$ that has parametric form $(x(t),y(t))$, so that $P=(x(a),y(a))$ and $Q=(x(b),y(b))$.

Then the length of the curve from $P$ to $Q$ is

$$L=\int_{a}^{b}\sqrt{\Bigl({{dx}\over{dt}}\Bigr)^{2}+\Bigl({{dy}\over{dt}}\Bigr)^{2}}\,dt.$$

We write

$$s(t)=\int_{a}^{t}\sqrt{\Bigl({{dx}\over{du}}\Bigr)^{2}+\Bigl({{dy}\over{du}}\Bigr)^{2}}\,du$$

for the arclength from $P$ to the point $(x(t),y(t))$, so

$$\Bigl({{ds}\over{dt}}\Bigr)^{2}=\Bigl({{dx}\over{dt}}\Bigr)^{2}+\Bigl({{dy}\over{dt}}\Bigr)^{2}.$$

The perimeter of a figure is the arclenth of the curve that goes once around the boundary.