### 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 $${\sin ^2}(t) + {\cos ^2}(t) = 1$$.

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

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

$L = \sqrt {10} (5 - ( - 5))$

$L = 10\sqrt {10}$