Answer
$$\frac{{3\sqrt 2 }}{4}{\text{m}}$$
Work Step by Step
$$\eqalign{
& {\text{The distance traveled is given by }} \cr
& s = \int_0^{\pi /8} {3\cos 2t} dt \cr
& {\text{Integrate and evaluate}} \cr
& s = \frac{3}{2}\left[ {\sin 2t} \right]_0^{\pi /8}{\text{m}} \cr
& s = \frac{3}{2}\left[ {\sin 2\left( {\frac{\pi }{8}} \right) - \sin 2\left( 0 \right)} \right]{\text{m}} \cr
& {\text{Simplify}} \cr
& s = \frac{3}{2}\left[ {\sin \left( {\frac{\pi }{4}} \right) - \sin \left( 0 \right)} \right]{\text{m}} \cr
& s = \frac{3}{2}\left[ {\frac{{\sqrt 2 }}{2}} \right]{\text{m}} \cr
& s = \frac{{3\sqrt 2 }}{4}{\text{m}} \cr} $$
