In order to find the distance traveled by a particle when given the velocity function of the particle, you take the integral of the absolute value of the velocity function. This means that the absolute value of the area under the curve of the velocity function equals the distance the particle travels. If the absolute value of the velocity function is not used, then the integral will be the displacement of a particle.

An example of calculating the distance traveled by a particle is a particle whose velocity function = 4. So:

               = 4    which means that the particle is moving at 4 units per time period 

The area under the curve from point A to point B is the distance that the particle travels for that time period.

This is because

If A = 0 and B = 3 then the particle traveled 12 units over the 3 units of time

The way to measure area under a curve is to use the integral of the function. Therefore, the distance traveled from time A to B for a particle whose velocity is modeled by the V(x)  =  (with the V(x) inside the integral being the absolute value)

A practical use for this concept is finding the distance traveled by a car if you know its speed.

How far does a car travel if it goes 70mph for 3 hours? Obviously the answer is 210 miles but let's practice using integrals.

The initial time = 0, the final time = 3, and V(x) = 70    so    = 210

A more complex problem is determining the distance a car travels while accelerating from rest. In this case V(x) = at  where a = acceleration and t = time

Find out how far a car travels if it accelerates at .001 miles per second for 5 seconds?

= .0125 miles or 66 feet

Here's a website with an applet to help understand graphicall the idea of distance traveled by a particle. Adjust the settings and think about how the area under the velocity curve corresponds with the distance traveled.

http://phet.colorado.edu/simulations/sims.php?sim=The_Moving_Man

Here are some multiple choice problems: (answers at bottom of page)

1.)  How far does a particle travel from time 3 to time 8 if its velocity is modeled by the function V(x) = 2 + .5t

     a.) 2.5

     b.) 23.75

     c.) 9.5

     d.) 16.25

2.) A particle moves along the x-axis so that at any time t>0 its velocity is given by v(t) = cos(t+t^(1/2)). The total distance traveled by the particle from t=0 to t=4 is         

          a.) 2.26

          b.) 2.30

          c.) 2.34

          d.) 2.38

          e.) 2.42

3.) A particle moves along a line in such a way that at time t, 1 < t < 8, its position is given by

          s(t) =

(a) Write a formula for the velocity of the particle at time t.

(b) At what instant does the particle reach its maximum speed?

(c) When is the particle moving to the left?

(d) Find the total distance traveled by the particle from t=1 to t=8.

answers: (b, e)

answers for free response:

a: v(t)= 1- t(cos t) - ((ln t)(sin t))

b: max |v(t)| = 5.896 when t = 6.700

c: 5.204 < t < 7.987

d: 21.461

Problem # 2 from "Preparing for the AP Calculus (BC) Examination" By George W. Best and J. Richard Lux

Some background information from "Calculus from Graphical, Numerical, and Symbolic Points of View, Single Variable" second edition by Ostebee and Zorn

Thomas Drew