We have seen that if v=v(t) is a function that represents the velocity of an object in motion from t=a to t=b, then
tells us that the distance travelled is equal to 
,
where 
is the velocity.
between t=a and t=b, but
not necessarily equal to , then the distance travelled is
approximately equal to .
, the
velocity cannot change too much, so that the distance covered over
this time interval is approximately equal to 
.
Notice that in each case, the product 
is
the area of a rectangle whose base lies on the t-axis and whose
height is given by the v coordinate of a point on the graph of
v=v(t). Intuitively, it appears that when the velocity varies
over the time interval t=a to t=b, we can approximate
the distance travelled from t=a to t=b by summing the areas
of rectangles that ``fill'' the region under the graph of v=v(t)
between a and b. This picture leads us to conclude that
this calculation is equivalent to ``finding the area under the
curve.'' Consequently, we'll use the language of area in the
rest of the discussion.