Introduction to Integration

In the previous examples, the function represented a rate of travel (miles per hour), and the area represented the total distance traveled. For functions representing other rates such as the production of a factory (bicycles per day), or the flow of water in a river (gallons per minute) or traffic over a bridge (cars per minute), or the spread of a disease (newly sick people per week), the area will still represent the total amount of something.

"Area" as a Total Accumulation

If $\mathrm{f}(t)$ represents a positive rate (in units per time interval) at time $t$, then the "area" between the graph of f and the t–axis and the vertical lines at times $t=\mathrm{a}$ and $t=\mathrm{b}$ will be the total units which accumulate between times $a$ and $b$. (Fig. 17)

For example, Fig. 18 shows the flow rate (cubic feet per second) of water in the Skykomish river at the town of Goldbar in Washington state. The area of the shaded region represents the total volume (cubic feet) of water flowing past the town during the month of October:

Total water = "area"

= area of rectangle + area of the triangle

$\approx(2000 \text { cubic feet } / \sec )(30 \text { days })+\frac{1}{2}(1500 \mathrm{cf} / \mathrm{s})(30 \text { days })=(2750 \text { cubic feet } / \mathrm{sec})(30 \text { days })$

$=(2750 \text { cubic feet } / \mathrm{sec})(2,592,000 \mathrm{sec}) \approx 7.128 \times 10^{9} \text { cubic feet }$

(For comparison, the flow over Niagara Falls is about $2.12 \times 10^{5} \mathrm{cf} / \mathrm{s}$).

Practice 9: Fig. 19 shows the number of telephone calls made per hour on a Tuesday. Approximately how many calls were made between 9 pm and 11 pm?