Exploring Accumulations of Change

A speed graph shows how fast something changes at every instant, but the question you usually want answered is how much changed in total. Area answers that. If velocity were constant, distance is just rate times time, and that product is exactly the area of a rectangle with height rate and width time. A curving velocity graph is no different in spirit: it just won't hold still long enough to be one rectangle.

So you slice the time interval into many thin strips. On each strip the rate barely changes, so the strip's area, height times width, is a good estimate of the amount accumulated during that sliver of time. Add up all the slivers and you get an estimate of the total change. Make the slivers thinner and more numerous, and the estimate becomes exact: the area under the whole curve equals the whole accumulated amount. This is why the units of area under a rate graph are not the same as the units of the rate itself. The vertical axis is measured in units like miles per hour, and the horizontal axis in hours. Multiplying them gives miles, the hours cancel out of the rate's units entirely, leaving a plain amount behind, not a rate.

This accumulation process is the reverse of what Unit 2 did. There, you took a quantity like position and zoomed in until the slope became an instantaneous rate. Here, you start with that rate and build back up to the total amount. Differentiation breaks a quantity down into its instantaneous rate; accumulation reassembles the rate into the quantity. That reversal is the heart of the Fundamental Theorem of Calculus, coming next.