Integration

Adding up infinitely many infinitesimally small pieces.

The idea

An integral is a sum — of infinitely many, infinitesimally small pieces.

To illustrate what integrals are about, suppose you are driving for an hour in a city. How far did you travel? If you drove in perfect conditions, on a highway with no traffic and no stops, the answer is easy: multiply your speed by the time. For example, if you go 60 mph for 1 hour, you will travel 60 miles. Easy.

But what if you slow down for traffic lights, you speed up to get through an intersection, you brake for pedestrians, then you accelerate again? Here is something you can do: glance at the speedometer once a minute, and pretend that your speed stayed constant for that minute, then add up the distances traveled in each minute. To get an even more accurate estimate, you can check the speedometer every second, or better yet every millisecond, and so on.

An integral captures the limit of this approximation as the time interval approaches zero. If at time $t$ we drive at speed $v(t)$, then the distance traveled between time $a$ and $b$ is given by $\int_{a}^{b} v(t) \, dt$, which we read as the integral from $a$ to $b$ of $v(t)$ with respect to $t$. Formally, this expression is defined to mean

$\displaystyle\int_a^b v(t) \, dt = \lim_{\Delta t \to 0} \sum v(t_i) \cdot \Delta t$

Ways to work on it

• Walkthrough. Estimate a distance by slicing, then squeeze the slices.
• Practice. Build a slice sum and judge which way it errs.
• Hardest. Why the limit does not depend on where you sample.

Not sure where to start? Take the ten-question placement test.