Arc Length

We can deduce that the length of a curve with 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 \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 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=b} is:

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 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:

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 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 Problems

1. 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}}

2. Find the length of the curve 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 y=\frac{e^x+e^{-x}}{2}} from 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=0} to 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=1} .

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_0^1\sqrt{1+\left(\frac{d}{dx}\left(\frac{e^{x}+e^{-x}}{2}\right)\right)^2}dx\\ &=\int\limits_0^1\sqrt{1+\left(\frac{e^{x}-e^{-x}}{2}\right)^2}dx\\ &=\int\limits_0^1\sqrt{1+\frac{e^{2x}-2+e^{-2x}}{4}}dx\\ &=\int\limits_0^1\sqrt{\frac{e^{2x}+2+e^{-2x}}{4}}dx\\ &=\int\limits_0^1\sqrt{\left(\frac{e^{x}+e^{-x}}{2}\right)^2}dx\\ &=\int\limits_0^1\frac{e^{x}+e^{-x}}{2}dx\\ &=\frac{e^{x}-e^{-x}}{2}\bigg|_0^1\\ &=\mathbf{\frac{e-\frac1e}{2}}\end{align}}

Resources

• Arc Length and Curvature, OpenStax