3.9 Area Under the Curve

Sometimes, we can derive a mathematical model by computing the area under the curve. The graph below is a velocity vs. time graph. Suppose an object traveled away from a fixed point at a constant 2.00 m/s for 6.00 s without changing speed or direction. If we collected data every one second, the data table and graph below would represent this event

The mathematical analysis follows:

The graph tells us there is no relation between velocity and time, because the graph is a horizontal line. The rate of change of velocity to time is zero. The equation represents a special case for a linear relation. Such an equation is not useful for any case other than the one shown above.

We can find a general model from this graph, however. If you observe the shaded area, from 0 to 6 seconds, you will see a rectangle. The shaded area has two pair of congruent sides and four right angles, making it a rectangle by definition. The mathematical model for the area of a rectangle is A = bh, where A is area of the rectangle, b is the base and h is the height.

The math analysis for the area under the curve follows:

The mathematical model d = v·t may be used to calculate the distance that any object travels as long as it is moving at a constant speed in a straight line path.