Doing this problem that involves distance a satellite travels with respect to earth (in thousands of miles). Find the total distance traveled from 0 to 5 hours.

$\displaystyle s(t) = t^3 - 2t^2 - 4T + 12 (position)$

$\displaystyle s'(t) = 3t^2 -4t -4 $

Critical Points: [-2/3, 2]

Now I am trying to gain a more tacit understanding of calculus, so my question is:

why is the integration of the velocity (position) from 0 to 5 hours

**not equal to** the sum of integrals between [0,2] and [2,5]? I thought that integration is summation, so why is it then when an object is traveling away from earth, you have to do a separate integral for it?

The answer is 71,000 miles, but I can't seem to comprehend why integral doesn't count the "negative" velocity in total distance, but the area summation does.

Sorry if this question is confusing, appreciate any help

.

-Warren