Your Electricity Bill Is an Integral

A smart meter samples your home's power draw and adds up power times time for every interval. That sum of thin rectangles is a Riemann sum, and as the intervals shrink it becomes the definite integral: the kilowatt-hours on your bill.

How it works

The definite integral and Riemann sums

In one sentence

If $f$ is continuous on $[a, b]$ , the Riemann sums $\sum f(x_i^*)\Delta x$ converge as $\max\Delta x \to 0$ , and the limit is the definite integral $\int_a^b f(x)\,dx$ .

General formula

{\htmlData{sym=0}{E}} = \int_0^{24} {\htmlData{sym=1}{P(t)}}\,dt = \lim_{{\htmlData{sym=4}{n}}\to\infty}\sum_{i=1}^{\htmlData{sym=4}{{n}}} P({\htmlData{sym=3}{t_i^*}})\,{\htmlData{sym=2}{\Delta t}}

Rate times time is amount. Chop time finely enough and the rate is constant on each piece, so add up the rectangles.

What the symbols mean

$E$
Energy
Kilowatt-hours used in the day.
$P(t)$
Power draw
Kilowatts the home uses at hour t.
$\Delta t$
Interval length
Hours between meter readings.
$t_i^*$
Sample time
The moment within each interval where power is read (the midpoint here).
$n$
Number of readings
How many intervals the day is split into.

Split the day into $n$ intervals of length $\Delta t = 24/n$ hours.
On each, power is nearly constant, so energy ≈ $P(t_i^*)\Delta t$ : a rectangle's area.
Adding the rectangles gives a Riemann sum; more, thinner rectangles get closer to the true energy.
The limit is the definite integral, the exact area under the power curve, in kWh.

Learn the math

The Area ProblemCalc I · Lesson 44, Integrals The Definite IntegralCalc I · Lesson 45, Integrals