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 : v(t) = t2 – 5t + 6 where v is measured in...

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 :

v(t) = t2 – 5t + 6

where v is measured in kilometres per hour and t is measured in hours.

Evaluate integral t=0 to t=4    (t2 – 5t + 6). What does the answer to this integral represent?

Asked on by adolphina

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

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.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll calculate the definite integral of the function that describes the speed: v(t) = t^2 - 5t + 6, using the Leibniz-Newton formula.

Int v(t)dt = Int (t^2 - 5t + 6)dt

Int (t^2 - 5t + 6)dt = Int (t^2)dt - 5Int t dt  + 6Int dt

Int v(t)dt = t^3/3 - 5t^2/2 + 6t 

Leibniz-Newton formula:

Int v(t)dt = V(4) - V(0), where V(t) = t^3/3 - 5t^2/2 + 6t.

V(4) = 64/3 - 80/2 + 24

V(4) = 64/3 - 40 + 24

V(4) = 64/3 - 16

V(4) = (64-48)/3

V(4) = 16/3

V(0) = 0

Int v(t)dt = 16/3

The answer represents the value of the area  enclosed by the curve of the speed function and the lines t = 0 and t = 4.

The area is A = 16/3 square units

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