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## Homework Statement

$$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$

## Homework Equations

Below

## The Attempt at a Solution

$$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$

I don't understand, we say:

$$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$

Then we say:

$$I = \int_{-\infty}^{\infty} e^{-t^2} dt$$

I want to see how this process works?

We consider

$$f(x) = e^{-x^2}$$

Right? Then we say that:

$$f(t) = e^{-t^2}$$

Right? Ideally we are replacing x with t correct? Then we say: where R is the whole

**real axis.**

$$I^2 = \int_{R} \int_{R} e^{-(t^2 + x^2)} dtdx$$

But the problem is that now you consider t and x different axes, while before they lay on the same axis.

How does this work?