From How the product of two integrals is iterated integral? $\int\cdot \int = \iint$ , the product of $\int f(x)\,dx$ and $\int g(y)\,dy$ is $\iint f(x)\,g(y)\,dx\,dy$.
From this, is the following true?
$$\left[\int_{a}^{b}f(x)\,dx\right]\left[\int_{a}^{b}g(x)\,dx\right] = \int_{a}^{b}\int_{a}^{b}f(x)\,g(x)\,dx\,dx$$
Please show your work so I can understand.
In general no, the identity $$ \left[\int_{a}^{b}f(x)\,dx\right]\left[\int_{a}^{b}g(x)\,dx\right] = \int_{a}^{b}\int_{a}^{b}f(x)\,g(x)\,dx\,dx $$ does not hold. As a counterexample, take $a=-1$, $b=1$ and $f(x)=g(x)=x$. Then your identity says $0=4/3$.
By the way, there should be no circumstances under which you find yourself constrained to use the same dummy variable (here $x$) more than once as in $\mathrm d x \mathrm d x$ (that is unless you run out of letters) because it creates ambiguity for you and your readers.