What does this integral notation mean?

[Leo Liu]

Homework Statement

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Relevant Equations

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I saw it somewhere but I did't know exactly what it meant. Could someone explain it to me like I am 5? Does it mean we integrate with respect to x n times?
$$\int_{\mathbb{R}^n}f\, \mathrm{d}^n x$$

 

[fresh_42]

Homework Statement:: .
Relevant Equations:: .

I saw it somewhere but I did't know exactly what it meant. Could someone explain it to me like I am 5? Does it mean we integrate with respect to x twice?
$$\int_{\mathbb{R}^n}f\, \mathrm{d}^n x$$

It means $\int_\mathbb{R}\int_\mathbb{R}\ldots\int_\mathbb{R}f \,dx_1\,dx_2\ldots\,dx_n$

 

[Leo Liu]

It means $\int_\mathbb{R}\int_\mathbb{R}\ldots\int_\mathbb{R}f \,dx_1\,dx_2\ldots\,dx_n$

Thanks. Just need some clarification -- do x-n s represent the same parameter or different variables?

 

[Mark44]

Thanks. Just need some clarification -- do x-n s represent the same parameter or different variables?

Different variables. The integral is over a region in $\mathbb R^n$. Each $dx_i$ is a different dummy variable, much the same as $\int \int f(x, y) dx dy$.