How do I convince my students that the choice of variable of integration is irrelevant? I will be TA this semester for the second course on Calculus, which contains the definite integral.
I have thought this since the time I took this course, so how do I convince my students that for a definite integral 
$$\int_a^b f(x)\ dx=\int_a^b f(z)\ dz=\int_a^b f(☺)\ d☺$$
i.e. The choice of variable of integration is irrelevant? 
I still do not have an answer to this question, so I would really hope someone would guide me along, or share your thoughts. (through comments of course)
NEW EDIT: I've found a relevant example from before, that will probably confuse most new students. And also give new insights to this question.
Example: If $f$ is continuous, prove that 
$$\int_0^{\pi/2}f(\cos x)\ dx = \int_0^{\pi/2}f(\sin x)\ dx$$
And so I start proving...
Note that $\cos x=\sin(\frac{\pi}{2} -x)$ and that $f$ is continuous, the integral is well-defined and
$$\int_0^{\pi/2}f(\cos x)\ dx=\int_0^{\pi/2}f(\sin(\frac{\pi}{2}-x))\ dx $$
Applying the substitution $u=\frac{\pi}{2} -x$, we obtain $dx =-du$ and hence
$$\int_0^{\pi/2}f(\sin(\frac{\pi}{2}-x))\ dx=-\int_{\pi/2}^{0}f(\sin u)\ du=\int_0^{\pi/2}f(\sin u)\ du\color{red}{=\int_0^{\pi/2}f(\sin x)\ dx}$$
Where the red part is the replacement of the dummy variable. So now, students, or even some of my peers will ask: $u$ is now dependent on $x$, what now? Why is the replacement still valid?
For me, I guess I will still answer according to the best answer here (by Harald), but I would love to hear more comments about this.

Maybe it helps to investigate the wording: integration variable is just a fancy name for what we used to call placeholder in elementary school when we solved
 3 + _ = 5

and used an underscore or an empty box as the placeholder. Isn't it obvious then that the symbol (or variable name) cannot have an effect on the solution?
