area x-axis function velocity
When velocity is constant, the total distance traveled over the given time interval can be computed by multiplying velocity and time.  For example, travelling a steady 50 mph for 3 hours results in a total distance of 150 miles.  Plotting the velocity as a function of time yields a rectangle with height equal to the velocity and width equal to the time elapsed.  Therefore, the product of velocity and time also calculates the rectangular area under the (constant) velocity curve.  This connection between the area under a curve and distance traveled can be extended to any irregularly shaped region exhibiting a fluctuating velocity over a given time period. If f(x) represents speed as it varies over time, the distance traveled between the times represented by a and b is the area of the region between f(x) and the x-axis, between x = a and x = b.
