A calculus problem by Ραμών Αδάλια

Given the functions

$f\left( t \right) =\frac { 1 }{ 3 } { t }^{ 3 }$

$g\left( t \right) =\frac { \sqrt { 5 } }{ \sqrt { 2 } } { t }^{ 2 }$

$h\left( t \right) =5t$

where $t$ represents the time in seconds, and $f\left( t \right)$ , $g\left( t \right)$ and $h\left( t \right)$ the position in the 3 distinct axis, find the distance traveled in the first 5 seconds, $a$ . Input your answer as $\left\lfloor a \right\rfloor$ .

Notes: By distance traveled, we understand the length of the arc of the trajectory, not the displacement between two points. The initial time is 0 seconds. Thus, the interval of time is from 0 to 5 seconds. Units are trivial.