# Thread: length of a curve

1. ## length of a curve

Find the length of :

1) $r(t) = cos(t){\bold{i}}+ sin(t){\bold{j}} + ln(cos(t)){\bold{k}}$

2) $r(t) = {\bold{i}} + t^2{\bold{j}} + t^3{\bold{k}} \ \ \ 0 \leq t \leq 1$

so far I have:
1) $r'(t) = (-sin(t){\bold{i}} + cos(t){\bold{j}}+ tan(t){\bold{k}}$

$|r'(t)| = \sqrt{sin^2(t) + cos^2(t) + tan^2(t)} = \sqrt{1+tan^2(t)} = \sqrt{sec^2(t)} = sec(t)$

$\int^{\frac{\pi}{4}}_{0}sec(t) = \int^{\frac{\pi}{4}}_{0}sec(\frac{\pi}{4})$

2) $r'(t) = (0 + 2t + 3t^2) \ \ \ |r(t)|= \sqrt{4t^2 + 9t^4} = t \sqrt{4 + 9t^2}$

is 1) correct and for 2) I'm stuck.

2. $\int{t\sqrt{4+9t^{2}}}dt$

Let $u=4+9t^{2}, \;\ du=18tdt, \;\ \frac{1}{18}du=tdt$

$\frac{1}{18}\int{u^{\frac{1}{2}}}du$

Now, integrate and resub.