Integral in two dimensions

Questions
1

Reading a scientific article, I came across the following integral

\int_0^1 dx \int\dfrac{d^2\vec{k}}{16\pi^3} \psi(x,\vec{k})^2

So, I had a huge question: how to use Mathematica to do integral \int d^2\vec{k} ?

Can anyone help me with this? Thanks!

2

In the documentation https://reference.wolfram.com/language/ref/Integrate.html

in the Examples, Basic Examples, Scope, Nested Integrals, it shows

Compute the second antiderivative of a function

Integrate[a x^2+b x+c, x, x]//Expand

So try

Integrate[Ψ[x,k]^2/(16 Pi^3),k,k]
3

If it were an iterated integral, I would somehow expect (dk)^2 instead of d^2 k. I wonder if the meaning is a double integral, with k a vector of dimension 2, although this would not explain the d^2 k.

4

Yes, “k” is a vector…