Arc Length of a Curve (Example 1)

In arc length questions you'll always be given an equation for a curve as well as the domain for which the curve goes from. You just need to memorize the arc length formula and use it!

Find the length of the curve \(r(t) = \left\langle {t,3\cos (t),3\sin (t)} \right\rangle \) over the domain \( - 5 \le t \le 5\).

Arc length is given by the formula:

\[L = \int_{{t_1}}^{{t_2}} {\left| {r'(t)} \right|} dt\]

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

\[r'(t) = \left\langle {1, - 3\sin (t),3\cos (t)} \right\rangle \]

Substituting into L, using the boundaries on t, and finding the norm:

\[L = \int_{ - 5}^5 {\sqrt {{{(1)}^2} + {{( - 3\sin (t))}^2} + {{(3\cos (t))}^2}} } dt\]

\[L = \int_{ - 5}^5 {\sqrt {1 + 9{{\sin }^2}(t) + 9{{\cos }^2}(t)} } dt\]

And use the identity \({\sin ^2}(t) + {\cos ^2}(t) = 1\).

\[L = \int_{ - 5}^5 {\sqrt {1 + 9} } dt\]

\[L = \int_{ - 5}^5 {\sqrt {10} } dt\]

\[L = \sqrt {10} (5 - ( - 5))\]

\[L = 10\sqrt {10} \]

 

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