Sizu Riatto
Calculus · Chapter 19

Average Value

The integral is the total built from a rate; divide it by the width and you have the true average. The halfway point between the high and the low is usually not it.

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Opens on Chapter 19’s example: the day’s temperature T(t) = 60 + 12t − t², from 6 a.m. (t = 0) to 6 p.m. (t = 12).

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How it works

The definite integral of a rate from a to b is the total it builds: the limit of the slice sums, worked out here with Simpson's rule on thousands of thin slices. Divide by the width b − a and you get the average value: the steady level that would build the same total in the same time. Parts below zero count as minus, so the integral is the net change; the total moved counts every part as plus. The halfway point between the highest and lowest values only equals the average for a straight line.

From the book, Chapter 19. The day’s integral is 1,008 degree-hours, so the true average is 84°F; halfway between the high 96 and low 60 is 78. A car pulling away from a light at v = t²/3 covers 192 feet in 12 seconds, averaging 16 ft/s, not the 24 halfway between its start and end speeds. The heat pump uses 12 kWh ($1.92 at 16¢); the charger, 21.6 kWh ($3.46). The tank gains 100 gallons and loses 36: net 64, 136 moved. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.