In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
The velocity v(t)v(t)v(t), in mm s−1\text{mm s}^{-1}mm s−1, of a microscopic probe is modeled by the function
v(t)=12tcos(3t),0≤t≤π6 v(t) = 12t \cos(3t), \quad 0 \le t \le \frac{\pi}{6} v(t)=12tcos(3t),0≤t≤6πwhere t t\,t is the time in seconds since the measurement began.
Find the total displacement of the probe over the interval 0≤t≤π6\displaystyle 0 \le t \le \frac{\pi}{6}0≤t≤6π, giving your answer in simplest form.
