Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 518

Table 11.3 shows tabulated values for 6, 12, and 24 slices. For each slice,
t 0 refers to the starting time of the slice, v 0 is the velocity at the start of the
slice (computed according to Equation (11.13) as v 0 = t 0

× 32 ft/s 2 ), ∆t is
the duration of the slice, and ∆x is our approximation for the displacement
during the slice (computed according to Equation (11.14) as ∆x = v 0 ∆t).
Since each slice has a different initial velocity, we are accounting for
the fact that the velocity changes over the entire interval. (In fact, the
computation of the starting velocity for the slice is not an approximation—
it is exact.) However, since we ignore the change in velocity within a slice,
our answer is only an approximation. Taking more and more slices, we get
better and better approximations, although it's di cult to tell to what value
these approximations are converging. Let's look at the problem graphically
to see if we can gain some insight.
In Figure 11.10, each rectangle represents one time interval in our ap-
proximation. Notice that the distance traveled during an interval is the
same as the area of the corresponding rectangle:

(area of rectangle) = (width of rectangle) × (height of rectangle)
= (duration of slice) × (velocity used for slice)
= (displacement during slice).

Now we come to the key observation. As we increase the number of
slices, the total area of the rectangles becomes closer and closer to the
area of the triangle under the velocity curve. In the limit, if we take an
infinite number of rectangles, the two areas will be equal. Now, since total
displacement of the falling ball bearing is equal to the total area of the
rectangles, which is equal to the area under the curve, we are led to an
important discovery.

The distance traveled is equal to the area under the velocity curve.

We have come to this conclusion by using a limit argument very similar to
the one we made to define instantaneous velocity—we consider how a series
of approximations converges in the limit as the approximation error goes
to zero.
Notice that we have made no assumptions in this argument about v(t).
In the example at hand, it is a simple linear function, and the graph is
a straight line; however, you should be able to convince yourself that this