I've TAed and tutored calculus for years and of the hundreds of students I've interacted with, it is always a shock when I tell them to change the limits of integration when they do substitutions. When I tell them to do that, they are always confused and act like they never heard about that. I remember being taught that. I might think that one or two teachers might not cover it but it's standard in calculus. Even students in Calculus 3 are shocked.
Why do students not know this?
Some students are instead taught to change the substitution variable back into the original variable before evaluating the antiderivative at the bounds.
$$\int\limits_0^2 2x\cos(x^2) \;\mathrm{d}x = \int\limits_{x=0}^{x=2} \cos(u) \;\mathrm{d}u = \sin(u) \Big|_{x=0}^{x=2} = \sin(x^2) \Big|_{x=0}^{x=2} = \sin(4) \\\text{versus}\\ \int\limits_0^2 2x\cos(x^2) \;\mathrm{d}x = \int\limits_{u=0}^{u=4} \cos(u) \;\mathrm{d}u = \sin(u) \Big|_{u=0}^{u=4} = \sin(4) $$
I recall doing it the former way the first time I was taught calculus, so I'm not sure how standard changing the bounds is. I think whether or not an instructor does this or instead changes the bounds is just a matter of preference (although I do worry that instructors gloss over the fact that if you don't change the bounds that they are still $x$-bounds, and so students don't really appreciate that and just change the variable back to $x$ because that's the procedure they've been taught). Now whether it is better to teach one or the other, or whether we should teach both, may be an interesting discussion.
Zach:
It is a standard part of how the subject is taught and is in every textbook that I have seen. It's also an easy thing to miss. Unless the students are well drilled, they won't internalize it.
Instead of being exasperated, just see it as one of those little things that people need reminding of. (Look at first year rookies in the NFL who forget you have to be "touched down" unlike the college game. Even though told it, they don't always internalize it.)
Also, realize that you as a TA are in the right end of the bell curve of calculus prowess. Your trainees will generally not be as strong as you are. Have some empathy and just help them get better. Don't be surprised they are not already better.
Nothing has changed. It was always a tricky thing and will always be. Every new batch of students brings new ignorance. That is how teaching and coaching are.
My guess is this may be due to practicing indefinite integration before definite integration, where such a thing does not come up. I imagine most people learn definite integration as "indefinite integration, followed by plugging in numbers", which lacks a substitution step.
At my institution (and I think this is common in semester systems, including the calculus AP curriculum), substitution of definite integrals is the very last subject of the first semester of calculus, so it's both the thing most likely to get shortchanged if the course runs behind schedule, and there's just less time to practice. When students pick up in the next semester, they usually jump in with "methods of integration"---integration by parts, trig substitution, and partial fractions---which reinforce the method of u-substitution, but are less likely to reinforce thinking about the bounds in definite integrals.
I suspect that students don't learn to change bounds of integration well because they have no motivation to do so. Whether we agree with them or not, it is evidently simpler for them conceptually to evaluate a definite integral by first evaluating the corresponding indefinite integral and then applying the fundamental theorem to that result.
As a teacher, there are a couple of things that I do to combat this. First, I devise problems where the whole point of the problem is to change bounds of integration in the first place. For example:
Express $\displaystyle \int_{-2}^{3} f(2x+1)\,dx$ as a definite integral involving $f(x)$.
Or,
Evaluate $\displaystyle \int_0^{\pi} \sin\left(e^{\sin(x)}\right)\cos(x)\,dx$.
Second, I try to impress upon them that there are plenty of practical situations where this translation from one definite integral to another arises. For example, when exploring probability theory (a natural application of integration in Calc II), we often need to translate a normal integral to a standard normal integral. This amounts to showing that $$\int_a^b e^{-(x-\mu)^2/(2\sigma^2)} dx = \int_{(a+\mu)/\sigma}^{(b+\mu)/\sigma} e^{-x^2/2} dx.$$ Thus, a rule that many of them learn in elementary statistics arises as a change of bounds in a definite integral.
As another example, that's nice if you do numeric integration, you might show that $$\int_0^1 \frac{\sin(1/x)}{x} \, dx = \int_1^{\infty} \frac{\sin(x)}{x} \, dx.$$ The point is that the second integral is more palatable from the point of view of numerical integration (rather than from the point of view of symbolic integration, as calc students are accustomed to). The reason is that the higher derivatives of the integrand don't explode over the domain of integration the way they do for the first. You can even illustrate this using a little sage code:
print numerical_integral(sin(1/x)/x, 0,1)
print numerical_integral(sin(x)/x, 1,1000)
# Out:
# (-1.9426263726635902, 42.80850345186512)
# (0.6241500516015874, 5.034861416675085e-14)
The first computation is completely wrong and has an absurd error bound. The second computation is (not surprisingly) correct to three decimal places.
Note that, in all four of these examples, the integrand cannot be integrated symbolically; a change of bounds is the whole point.
Following on Mike's example, you might insist on explicitly specifying the variables being integrated over in the limits as well as the substitution used:
$\int\limits_{x=0}^{x=2} 2x\cos(x^2) \;\mathrm{d}x = \int\limits_{u=0}^{u=4} \cos(u) \;\mathrm{d}u = \sin(u) \Big|_{u=0}^{u=4} = \sin(4) \space\space$ where $u=x^2, \space du=2x\space dx$
I'm surprised this hasn't been mentioned yet, but it seems likely that the OPs experience is due in part to selection bias. That is: the students that one encounters when tutoring, or in office hours, are more likely to be precisely the students that are struggling to understand something. There are probably lots and lots of students who are learning this skill when it is taught to them; you aren't seeing them, because they don't need to come to you for help.
I found that it was safer for weaker students to evaluate the indefinite interval, back substitute, and the evaluate the limits once. Multiple evaluations just gave them additional additional steps in which to make errors and didn't really improve their somewhat limited grasp of what it all meant.
