t - Time elapsed
$\Delta$t - Time interval for approximating the position, by summing up discrete values, $v(t_{i}) \Delta t$
Things to Notice
Negative Velocity
Drag the time slider ($t$) past 4 seconds. Notice that the velocity curve dips below the horizontal axis, and the rectangles are drawn downward. The velocity values are negative, so the ball moves backwards. The integral treats areas below the x-axis as negative, and subtracts them instead of adding.