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.
Draw a graph of the function on the blackboard, showing $a$ and $b$ and a crosshatched area representing the integral. Put an $x$ on the horizontal axis. Erase the $x$ and put a $z$ there. Does that change the area? Erase the $z$ and put a smiley face there. Does the area change? Why/why not?
I think something to think of here is the fact that in teaching calculus (especially the definite integral; which from my experience is a second semester calculus class) the students should already be aware of the fact that x is a variable and the choice of letter (or symbol) is completely arbitrary. I would be more worried about the students that can't see this and would see it as them being in the wrong class. If they don't understand the fundamental basis of a variable they probably shouldn't be in a calculus class.

I would venture to say that this shouldn't be/isn't a problem for a university level calculus class.

If you truly are worried about this, I would suggest not focusing on the topic but instead showing it through examples; continually substitute in whimsical pictographs in place of the constant x/y/z. It would be entertaining to the students and would drive the point home without you actually having to focus on it.