Length Of Curve

#Math

Topics

• Arc Length

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

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

$\displaystyle L=\int_{\alpha}^{\beta} \sqrt{ \left( \frac{\mathrm{d}x }{ \mathrm{d}t} \right)^{2}+\left( \frac{\mathrm{d}y }{ \mathrm{d}t} \right)^{2} } , \mathrm{d}t=\int_{a}^{b} \left\lVert \vec{r}'(t)\right\rVert , \mathrm{d}t$

• For parametric equations