We can deduce that the length of a curve with parametric equations
,
should be:
![{\displaystyle L=\int _{a}^{b}{\sqrt {{\biggl (}{\frac {dx}{dt}}{\biggr )}^{2}+{\biggl (}{\frac {dy}{dt}}{\biggr )}^{2}}}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76976463e914bfb3e9649f9bf055cc4302ecc79e)
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
, or, equivalently, the parametric equations
, where
are continuous, then the length of the curve from
to
is:
}}
For those who prefer simplicity, the formula can be rewritten into:
or ![{\displaystyle \quad {\frac {dL}{dt}}=|\mathbf {r} '(t)|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7af8a9ad556637028c9716854b306112ad1b3037)
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