Length of Curve

Topics

• Arc Length

$L=\int \mathrm{d}s$

• $\mathrm{d}s=\sqrt{1+{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}^2}\mathrm{d}x$ if $y=f(x),a\le x\le b$
• $\mathrm{d}s=\sqrt{1+{\left(\frac{\mathrm{d}x}{\mathrm{d}y}\right)}^2}\mathrm{d}y$ if $x=h(y),c\le y\le d$
• From Paul’s Online Notes

$L={\int }_{\alpha }^{\beta }\sqrt{{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)}^2+{\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)}^2}\mathrm{d}t={\int }_a^b\Vert {\vec{r}}^{\prime }(t)\Vert \mathrm{d}t$

• For parametric equations