A calculus problem by Ραμών Αδάλια

Calculus Level pending

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.

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