I need to find the LENGTH of the curve with x,y,z components listed below:
$ x(t) = t\sin(2t)$
$ y(t) = t\cos(2t)$
$z(t) = (4/3)t\sqrt(t) = (4/3)(t^{1.5})$
from $t= 0$ to $t=2\pi$
can anyone help pls? i have tried but came to a very hard integral which doesnt look right
i get
$L =2 \int^{2\pi}_0\sqrt{t^2 + t}$
which looks wrong since the online solution to this is very lengthy.
You have to use the formula of the length of a curve: $\int ||\varphi|| dt$ from $0$ to $2\pi$. in this case the derivative of the components are: $$x'=\sin{(2t)} + 2t\cos{(2t)}$$ $$y'=\cos{(2t)}- 2t\sin{(2t)}$$ $$z'=2t^{0.5}$$
follows that the function as an integral is $2t+1$ because $2t+1$ is positive in $[0,2\pi]$. And now this integral is very easy.