Concept preview

What you will be able to do

1. 01Explain how a rate of change accumulates into a total, and recognise the accumulated change as the area between the rate graph and the horizontal axis.
2. 02Compute accumulated change in context — for example distance from a velocity graph or volume from a flow rate — from a rate given as a formula, a graph, or a table.
3. 03Distinguish signed accumulation (net change, where regions below the axis subtract) from total geometric area, and write accumulated change with the integral symbol $\displaystyle\int$.

Start with the simplest possible rate — a constant one. Suppose a tap pours water into a bucket at a steady $5$ litres per minute. After $8$ minutes the bucket holds $5\times 8 = 40$ litres. You multiplied a rate (litres per minute) by an elapsed time (minutes) to get a total (litres). Notice what that multiplication is geometrically: if you graph the rate $R=5$ as a flat horizontal line and mark off the time interval from $t=0$ to $t=8$ on the axis beneath it, the product $5\times 8$ is exactly the area of the rectangle trapped between that line and the axis. The accumulated total is the area under the rate graph. Hold on to that sentence — everything else is a variation on it.