The velocity of a particle moving along the x-axis is v(t) = t2 – 2t, with t measured in minutes and v(t) measured in feet per minute.

Question

To the nearest foot find the total distance traveled by the particle from t = 0 to t = 3 minutes.

I'm pretty sure it's just the integral from 0 to 3 of the above equation, which is 0, but I'd rather be safe than sorry.

Thanks in advance.

Answer 1

Hi, when you're looking for total distance traveled you must take into account whether or not the velocity function's graph is above the x axis. If you graph:

v(t)=t^2-2t

you'll see that from 0 to 2 the function is below the x axis and from 2 to 3 the function is above it. While the function is below the x axis your result will be negative and while above the result will be positive. You'll have to separate the integral from 0 to 3 into the integrals from 0 to 2 and from 2 to 3. Because the function is below the x axis from 0 to 2 you'll have to put a negative in front of this integral to ensure you get a positive value for it and adding the results of these two integrals will yield the correct answer.