We give an alternative interpretation of the definite integral and make a connection between areas and antiderivatives.

We have a geometric interpretation of the derivative as the slope of a tangent line at a point. We have not yet found a geometric interpretation of antiderivatives.

More than one perspective

We’ll start with a question:

Suppose you are in slow traffic moving at mph from pm to pm. How far have you traveled?
Since the displacement of an object moving at constant velocity over a time interval can be represented by the area of a rectangle, we can approximate displacement of an object moving at nonconstant velocity using Riemann sums. We explain this process in detail in our next example.