One of the most useful mathematical insights I gained from working as an astronomer is that the bounds in an integral are inextricably linked to the variable of integration, and that this plays an absolutely critical role in integration by substitution. Changing the bounds when you change the variable isn’t just a shortcut or a convenience: it is an essential part of the process of substitution. And it makes total sense if we look at substitution in a physical context in which we change between two variables that actually mean something to us. The problems linked here introduce substitution by placing it in a physical context and clarifying what it really means to change the variable of integration.
In most calculus books, substitution is described primarily as a way to simplify integrals, for example converting \(\scriptsize\int\sin(x^2)2x{\rm d}x\) into the simpler form \(\scriptsize\int\sin(u){\rm d}u\). However, to physicists it is at least as important to regard substitution as a tool for changing the variable of integration, and this may be an end unto itself. For example, suppose we are talking about the change in air pressure an airplane experiences as it rises. If we ask “How much does the pressure change from time a to time b,” we might use an integral taken with respect to time, t. On the other hand, if we ask “How much does the pressure change from elevation a to elevation b,” then we want an integral taken with respect to position, x. Substitution is the tool that lets us switch between integrating over time and integrating over position. You might want to switch between the two, not because one form is simpler, but because the integrals mean different things and a particular problem may require one or the other.
In this case, converting \(\scriptsize{\rm d}x\) into \(\scriptsize{\rm d}t\) has an extremely clear meaning, and it is clear that this conversion requires us to change the bounds in a definite integral. The change in pressure from position a to position b is simply a different thing than the change in pressure from time a to time b. If \(\scriptsize x(a)\) and \(\scriptsize x(b)\) are the positions corresponding to times a and b, then the change in pressure from time a to time b is the same as the change in pressure from position \(\scriptsize x(a)\) to position \(\scriptsize x(b)\): we must use the positions corresponding to the given times. And so an integral \(\scriptsize {\rm d}t\) with bounds a and b can only ever be compared to an integral \(\scriptsize {\rm d}x\) with bounds \(\scriptsize x(a)\) and \(\scriptsize x(b)\).
The problems linked here use this sort of physical intuition to introduce substitution. The conceptual framework for these problems is introduced in these chain rule problems and will need a little introduction if you haven’t done those problems (here is a quick-start guide to those chain rule problems if you want to use them). The idea is that someone is walking up a hill, and that their elevation E depends on their position x, which depends on time t:
In this context, $$\int\frac{{\rm d}E}{{\rm d}t}{{\rm d}t} \text{ and } \int\frac{{\rm d}E}{{\rm d}t}{{\rm d}t}$$ represent changes in elevation over a range of time and over a range of position, respectively. And for these to be equal, we need the range of position to correspond to the given range of time:$$\int_a^b\frac{{\rm d}E}{{\rm d}t}{{\rm d}t} =\int_{x(a)}^{x(b)}\frac{{\rm d}E}{{\rm d}t}{{\rm d}t}.$$ The integral on the left gives us a gain in elevation from time a to time b, whereas the integral on the right gives us the gain in elevation from the person’s location at time a to their location at time b. But these two things are the same: the change in elevation between two times is precisely the change in elevation between the locations that go with those times.
If we use the chain rule to replace \(\scriptsize \frac{{\rm d}E}{{\rm d}t}\) with \(\scriptsize \frac{{\rm d}E}{{\rm d}x}\frac{{\rm d}x}{{\rm d}t}\), then we are doing integration by substitution. The idea of the problems linked in this post is to start with this physical framework and use it to introduce substitution as a more abstract concept and a way of simplifying integrals. If you start with those problems and it goes well, the other problems in this folder serve to extend the same framework to introduce substitution on increasingly complex integrals.
None of this proves that substitution works, but it provides an intuitive physical framework for understanding what it does. It is meant to complement whatever proof you may give of substitution and the chain rule.
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