find the length of the curve r(t) = <2t, t^2, 1/3t^3>
I did this
r'(t) = <2, 2t, t^2>
|r'(t)|
When I take the integral, I don't know how to solve for it.
Thanks in advanced.
First, what are the boundaries on t?
Second, the integral is $\displaystyle \int_{t_0}^{t_f}\sqrt{\left(\frac{\,dx}{\,dt}\righ t)^2+\left(\frac{\,dy}{\,dt}\right)^2+\left(\frac{ \,dz}{\,dt}\right)^2}\,dt=\int_{t_0}^{t_f}\sqrt{4+ 4t^2+t^4}\,dt=\int_{t_0}^{t_f}\sqrt{(t^2+2)^2}\,dt$ $\displaystyle =\int_{t_0}^{t_f}\left(t^2+2\right)\,dt$, for $\displaystyle t_0\leq t\leq t_f$
This is not much of a hassle to integrate.
Does this make sense?
--Chris