Answer
At time $T$, the bee is located at the origin if $\mathop \smallint \limits_0^T {\bf{r}}'\left( u \right){\rm{d}}u = {\bf{0}}$.
The quantity $\mathop \smallint \limits_0^T ||{\bf{r}}'\left( u \right)||{\rm{d}}u$ represents the length of the path traced by the bee for the interval $0 \le t \le T$.
Work Step by Step
Recall the position vector is given by
${\bf{r}}\left( t \right) = \smallint {\bf{r}}'\left( u \right){\rm{d}}u$
So, $\mathop \smallint \limits_0^T {\bf{r}}'\left( u \right){\rm{d}}u = {\bf{0}}$ implies that the position vector after traveling $T$ time is located at the origin.
Whereas, $\mathop \smallint \limits_0^T ||{\bf{r}}'\left( u \right)||{\rm{d}}u$ represents the length of the path traced by the bee for the interval $0 \le t \le T$.
