Finding the distance traveled over a length of time (of an object)
Questions 1c and 2 on the review test as us to find an integral that could be solved in order to find the distance traveled by an object over a parametric plot. I can visualize this process, however, I am not quite sure how to go about setting up an integral for this using both \vec{x} = t + cos(4t) and \vec{y} = -sin(4t).
Here question 1c is asking us to find the integral described above for the following vector
$$ \vec{p}(t) = \langle t + cos(4t), -sin(4t) \rangle $$ and $$ \vec{p'}(t) = \langle 1 - 4sin(4t), -4cos(4t) \rangle $$
I believe setting up the integral would set up using the derivative of the position function integrated over the time interval giving the integral
$$ \int_0^{2\pi} \ <1 - 4sin(4t), -4cos(4t)> \mathrm{d}t $$
Any help would be appreciated