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
∫baf(x) dx=∫baf(z) dz=∫baf(☺) 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
∫π/20f(cosx) dx=∫π/20f(sinx) dx
And so I start proving…
Note that cosx=sin(π2−x) and that f is continuous, the integral is well-defined and
∫π/20f(cosx) dx=∫π/20f(sin(π2−x)) dx
Applying the substitution u=π2−x, we obtain dx=−du and hence
∫π/20f(sin(π2−x)) dx=−∫0π/2f(sinu) du=∫π/20f(sinu) du=∫π/20f(sinx) 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?