Question

Use of numerical integral method to calculate the length of the path has traversed the car during the time recording below

Answer 1

Don't you have more informations concerning the method of interpolation of the velocities ? Is this linear interpolation ?

Answer 2

Well, assuming linear interpolation of velocities, you can write:
\[V(t) = V0 + (V1-V0) t\]\[D(t) = \int\limits\limits_{0}^{t}v(x).x.dx\]
Here, V0 and V1 are 2 consecutive values in your array. Like 124 and 134 for the first 2 values. D(t) is the distance you went through during time t. For this particular example of yours, we're going to always use t=6 because your time steps are constant and always equal 6 seconds.

I used http://integrals.wolfram.com/index.jsp?expr=a+x+%2B+b+x%5E2&random=false to perform the integral (although it's very easy to write it yourself) so we find that:
\[D(t) = \int\limits\limits_{0}^{t}v(x).x.dx\ = \left[ \frac{ V0 }{ 2 } t^2 + \frac{ V1-V0 }{ 18 } t ^{3} \right]\]
When solving for D(6) (t varying in [0,6]) we get:
\[D(6) = 18 V0 + 12(V1-V0) = 12 V1 + 6 V0\]

All you need to do next is to write a loop that sums \[12 V1 + 6 V0\] for all you time intervals. It goes like:
float SumDistance = 0;
for ( int TimeStepIndex=0; TimeStepIndex < TimeValuesCount-1; TimeStepIndex++ )
{
float TimeStepDistance = 6 * Velocities[TimeStepIndex] + 12 * Velocities[TimeStepIndex+1];
SumDistance += TimeStepDistance;
}

Answer 3

Sorry, my first equation should read:
\[V(t) = V0 + \frac{ V1 - V0 }{ 6 } t\]
because starting from V0 at t=0 second, we only reach V1 after 6 seconds...

Answer 4

If the actual assignment was to perform numerical integration using many small time steps then I suppose my solution is wrong, instead you should go all code and write something like:

int IntegrationStepsCount = 1000; // The more you use, the more precise it will get, but never as precise as an analytical integration obviously...
float SumDistance = 0;
for ( int TimeStepIndex=0; TimeStepIndex < TimeValuesCount-1; TimeStepIndex++ )
{
float TimeStepDistance = 0;
float StartVelocity = Velocities[TimeStepIndex];
float EndVelocity = Velocities[TimeStepIndex+1];
float IntegrationStepSize = 6.0 / IntegrationStepsCount;
for ( int i=0; i < IntegrationStepsCount ; i++ )
{
float CurrentVelocity = StartVelocity + (EndVelocity - StartVelocity) * i / IntegrationStepsCount;
TimeStepDistance += CurrentVelocity * IntegrationStepSize; // Distance += Velocity * DeltaTime
}
SumDistance += TimeStepDistance;
}

This is a purely numerical integration...