The Fundamental Theorem of Calculus
Velocity is a vector, meaning it has both magnitude and direction. The "direction" of velocity is either positive or negative. Positive and negative velocities describe motion in opposite directions. In Dance Dance Revolution, negative would either be juking left or back and positive would be moving towards the forward and right arrows. However, if the player would rotate 90 degrees, the relative negative and positive directions might change. DDR would enter bazaar-o world, but it could happen. The context of a problem will tell you how to interpret positive and negative velocities for that problem.
1) Let v(t) be the velocity of a bug crawling back and forth on the x-axis. When v(t) is positive, the bug is moving to the right:
When v(t) is negative, the bug is moving to the left.
2) Calvin is biking away from his house with velocity v(t).
When v(t) is negative it means Calvin is biking back towards his house.
3) A bird is flying North with velocity v(t).
When v(t) is negative it means the bird is flying South.
Speed is the magnitude, or absolute value, of velocity. This means speed can't be negative!
Let v(t), in units per second, be the velocity of a bug crawling back and forth on the x-axis. When v(t) = -5 it means the bug is crawling left at a speed of 5 units per second.
When a question asks "how fast" something is going, you're being asked for the speed.
When Calvin's velocity is -7 miles per hour, how fast is he going?
The question is asking for Calvin's speed, which is 7 miles per hour. The negative sign tells which direction Calvin is traveling, but doesn't have anything to do with his speed.
Now that we know what negative velocities mean, it's time to bring the integrals into it again and figure out what integrals of negative velocities mean.
When velocity is non-negative, we know that
When velocity is negative, the integral of velocity is also negative. We can think of this negative value as a weighted distance.
The number part tells the distance, and the negative sign tells the direction, which is the opposite of the direction travelled when v(t) is positive.
Take the area between v(t) and the t-axis that's above the axis. This is the total distance the bug travels to the right.
Take the area that's between v(t) and the t-axis, below the t-axis. This is the total distance the bug travels to the left.
(distance bug travels right) – (distance bug travels left) = 9 – 19 = -10.
This is the net change in the bug's position from time t = 0 to time t = 10.
We found this net change by taking a weighted sum of the areas between v(t) and the t-axis, which means we also just found the integral of v(t) from 0 to 10:
Whatever the units, when velocity is 0, speed is also 0. If something isn't moving at all, it's not moving in any direction. If v(t) is Calvin's velocity away from his house, measured in mph, then when v(t) = 0 Calvin isn't moving away from his house and he's not moving towards his house either.