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

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(π2x) and that f is continuous, the integral is well-defined and

π/20f(cosx) dx=π/20f(sin(π2x)) dx

Applying the substitution u=π2x, we obtain dx=du and hence

π/20f(sin(π2x)) 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?

Source : Link , Question Author : Community , Answer Author :
Harald Hanche-Olsen

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