Arc Length

We can deduce that the length of a curve with parametric equations ${\begin{cases}x=f(t)\\y=g(t)\end{cases}}$, $a\leq t\leq b$ should be:

$L=\int _{a}^{b}{\sqrt {{\biggl (}{\frac {dx}{dt}}{\biggr )}^{2}+{\biggl (}{\frac {dy}{dt}}{\biggr )}^{2}}}dt$

Since vector functions are fundamentally parametric equations with directions, we can utilize the formula above into the length of a space curve.

Arc length of a space curve

If the curve has the vector equation $\mathbf {r} (t)=\langle f(t),g(t),h(t)\rangle ,a\leq t\leq b$, or, equivalently, the parametric equations $x=f(t),y=g(t),z=h(t)$, where $f',g',h'$ are continuous, then the length of the curve from $t=a$ to $t=b$ is:

$L=\int _{a}^{b}{\sqrt {[f'(t)]^{2}+[g'(t)]^{2}+[h'(t)]^{2}}}dt=\int _{a}^{b}{\sqrt {{\biggl (}{\frac {dx}{dt}}{\biggr )}^{2}+{\biggl (}{\frac {dy}{dt}}{\biggr )}^{2}+{\biggl (}{\frac {dx}{dz}}{\biggr )}^{2}}}dt$}}

For those who prefer simplicity, the formula can be rewritten into:

$L=\int _{a}^{b}|\mathbf {r} '(t)|dt\quad$ or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \quad\frac{dL}{dt}=|\mathbf{r}'(t)|}

Example Problem

Find the circumference of the circle given by the parametric equations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t)=R\cos(t),y(t)=R\sin(t)} , with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t\in[0,2\pi]} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}s&=\int\limits_0^{2\pi}\sqrt{\left(\tfrac{d}{dt}\big(R\cos(t)\big)\right)^2+\left(\tfrac{d}{dt}\big(R\sin(t)\big)\right)^2}dt\\ &=\int\limits_0^{2\pi}\sqrt{\big(-R\sin(t)\big)^2+\big(R\cos(t)\big)^2}dt\\ &=\int\limits_0^{2\pi}\sqrt{R^2\big(\sin^2(t)+\cos^2(t)\big)}dt\\ &=\int\limits_0^{2\pi}Rdt\\ &=R\cdot t\Big|_0^{2\pi}\\ &=\mathbf{2\pi R}\end{align}}

Resources

• Arc Length and Curvature, OpenStax