Substitution for definite integrals
In my experience, Calculus II students dislike changing bounds in definite integrals involving substitution. When facing an integral like $$\int_0^{\sqrt{\pi }} x \sin \left(x^2\right)dx,$$ for example, most US Calc II students would introduce $u=x^2$ and compute \begin{align} \int x \sin \left(x^2\right)dx &= \frac{1}{2}\int \sin(u) \, du \\ &= -\frac{1}{2} \cos(u)+c = -\frac{1}{2} \cos(x^2)+c. \end{align} Afterward, they would conclude that $$\int_0^{\sqrt{\pi }} x \sin \left(x^2\right) \, dx = -\frac{1}{2} \cos(x^2) \big|_0^{\sqrt{\pi}} = 1.$$ I would generally encourage them to write \begin{align} \int_0^{\sqrt{\pi }} x \sin \left(x^2\right)dx &= \frac{1}{2}\int_0^{\pi} \sin(u) \, du \\ &= -\frac{1}{2} \cos(u) \big|_0^{\pi} = 1. \end{align} This question expresses opinion of a typical such student and this answer correctly expresses the fact that the two step process favored my most calculus students is actually more work.
I think there's more to it than this, though. Specifically, the identity $$\int_a^b f(g(x)) g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du$$ is a relationship between definite integrals which could have applications other than symbolic evaluation of the integral on the left. In this case, the change of the bounds of integration is important in its own right. Thus my question:
What are some important applications of change of variables in definite integration, other than symbolic evaluation?
I have at least one answer but would be happy to hear more, particularly those that are easily understandable by Calc II students, as I think it's an important pedagogical question.
What are some important applications of change of variables in definite integration, other than symbolic evaluation?
Here are a handful:
To give a computational (as opposed to geometric) proof that $$ \int_{-a}^{a} f(x)\, dx = \begin{cases} 0 & \text{if $f$ is odd}; \\ 2\displaystyle\int_{0}^{a} f(x)\, dx & \text{if $f$ is even.} \end{cases} $$
To show that if $a$ and $b$ are positive real numbers, then $$ \int_{a}^{ab} \frac{1}{t}\, dt = \int_{1}^{b} \frac{1}{t}\, dt, $$ which, of course, is the key to proving $\log(ab) = \log(a) + \log(b)$.
To prove that if $f$ is continuous on $[-1, 1]$, then $$ \int_{0}^{2\pi} f(\cos \theta) \sin\theta\, d\theta = \int_{0}^{2\pi} f(\sin \theta) \cos\theta\, d\theta = 0. $$
Here's another common example: $$ \int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx.$$
The geometric interpretation is simply that the area of a region is preserved by reflection about its midsection.
