I have just been introduced to the topic of integration. My teacher said:   $$\int_{a}^{b} f(x) dx$$  Is the same thing as:  $$\int_{a}^{b} f(\alpha) d\alpha$$  I asked her why, and she said "it is a dummy variable."  But then I thought, are all variables dummy variables? how do you identify a dummy variable?
I do not understand how dummy variables work in math.   Suppose we have:  $$I_1 = \int_{0}^{\infty} e^{-x^2} dx$$  How is this equivalent to:  $$I_2 = \int_{0}^{\infty} e^{-y^2} dy$$  How does this dummy variable system work?   Since $y$ is the dependent variable for $I_1$ How can $y$ itself be and independent variable for $I_2$  ??  Thanks!
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