You usually hear about pullbacks in advanced calculus, but we’ve already snuck one by you in calc 1. Let me show you where.

Contents:

The setup

  1. Walk along the \(x\) axis.
  2. Stop for a sec every once in while and calculate \(f(x)\).
  3. Sum up the areas of some (red) rectangles.

  4. Do step 2. often enough and you get

    \[\int_{a}^{b}f(x)dx \approx \sum f(x_i)\Delta x_i.\]

Try it out live:

Going fast

Stopping every \(\Delta t_i\) seconds (this is the once in a while from step 2.) is really easy, just keep an eye on your watch. On the downside, there’s the side effect that the rectangle bases are uneven when we speed up or slow down. We can fix the aesthetics by bringing time front and center.

Somewhat surprisingly, the blue and the red rectangles give us different numbers. What went wrong? We slowed down at the top to look at the view, but the relevant blue rectangle doesn’t know we haven’t moved much during his \(\Delta t_i\) (Time waits for no one…). Conveniently, his red friend knows exactly how much we moved during that time, it’s his \(\Delta x_i\).

You might want to click a rectangle up there right now.

Let’s redo the sum with this new information.

This time, each blue rectangle reminded us how much \(x\) we moved during his \(\Delta t_i\). It momentarily expanded\contracted to the right size by multiplying its base by \(\frac{\Delta x_i}{\Delta t_i}\) and then snapped back once we were done with it. this gave us the right number:

\[\sum f(x(t_i))\frac{\Delta x_i}{\Delta t_i}\Delta t_i = \sum f(x_i)\Delta x_i.\]

At the limit \(\Delta t_i \to 0\) this becomes integration by substitution:

\[\int_{x^{-1}(a)}^{x^{-1}(b)}f(x(t))x^\prime (t)dt = \int_{a}^{b}f(x)dx.\]

Pullbacks galore

We moved geometric information (about lengths of \(\Delta x_i\)’s) from the world of \(x\) back to the world of \(t\). The jargon for this is pullback and it shows up everywhere once you start looking. Start looking here:

  • You’ve used a map with a scale, so now you know. In this case the map pulls back lengths from the actual world to the paper it’s printed on.

    Map Scale
    A pullback
  • For really good maps, this 1d scale doesn’t cut it anymore. You need to pull back a metric tensor. The first 10 mins of this video are a great intro, though it picks up speed afterwards.
  • Steve Mould has some other, physics inspired examples.

Reverse chain rule

Summing rectangles is cute, but you should check out the proof [Wiki]. It’s a one liner using the chain rule:

\[\frac{df}{dt}=\frac{df}{dx}\frac{dx}{dt}.\]

You understand this one intuitively if you’ve ever switched gears in a car. If not, check out the viz here.