Arc Length of a Curve - Reparametrization a Curve (Example 2)

Reparametrize the curve r(t) with respect to arc length, starting from t = 0 to a direction of increasing t, where \(r(t) = 2ti + (1 - 3t)j + (5 + 4t)k\).

Arc length is given by:

\[s(t) = L = \int_0^t {\left| {r'(u)} \right|} du\]

Note that the equation above is for r(t), but you're not technically allowed to have the same variable for r as what is in the integral boundaries (t). Even though it says r(u), it really is just r(t).

We start by finding the derivative of r(t).

\[r'(t) = 2i - 3j + 4k\]

Then substitute it into the arc length formula.

\[s(t) = \int_0^t {\sqrt {{{(2)}^2} + {{( - 3)}^2} + {{(4)}^2}} du} \]

\[s(t) = \sqrt {29} (t - 0)\]

\[s(t) = \sqrt {29} t\]

Rearrange in terms of t, and substitute it into r(t). That's just what arc length parametrization is.

\[t = \frac{s}{{\sqrt {29} }}\]

And our original r(t) is:

\[r(t) = 2ti + (1 - 3t)j + (5 + 4t)k\]

Substituting in we obtain:

\[r\left( {t(s)} \right) = \frac{{2s}}{{\sqrt {29} }}i + \left( {1 - \frac{{3s}}{{\sqrt {29} }}} \right)j + \left( {5 + \frac{{4s}}{{\sqrt {29} }}} \right)k\]

Done! Sometimes the final substitution gets messy arithmetically.

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