What does the integral of speed represent in the following case?The speed of a motorboat travelling in a straight line out at sea is given by :...

Since the boat is travelling in a straight line, let the
distance function travelled by the boat along the lineat time be s(t)
imles.

Thenthe instantaneous velocity of the boat at any
time t is given by  ds/dt = s'(t).

Therefore velocity s'(t)
= v(t) = t^2-5t+6 which is a function of time t in
hours.

Therefore s(t) = Int s'(t) dt = Int v(t) dt = Int
(t^2-5t+6) dt

Therefore s(t) =
Int{t^2-5t+6)dt.

s(t) = {t^3/3-5t^2/2+6t}
+C.

S(4)  = {4^3/3-5*4^2/2+6*4} +C = {64/3-5*16/2+6*4}+C =
16/3 = 16/3 +C miles of distance  from the beginning in 4
hours.

s(0) = (0^3/3-5*0^2/2+6*0}+C = C miles from the
beginning at the time of 0 hour.

s(4) -s(0) = 16/3+C- C =
16/3 miles traversed  by the boat in the interval of time from 0 to 4
hours.

Hope this helps.