Integral $\int_{[1,2]\times[0,\pi]}\log(\sqrt{x})\sin(2y)d(x,y)$

Question

I want to find out if this integral can be calculated (if it exists)

$$\int_{[1,2]\times[0,\pi]} \log(\sqrt{x})\sin(2y)~d(x,y)$$

To be honest, I don't know how, but I think that one might has to use Fubini's theorem since this is an iterated integral. Does someone know how it's done? And can someone explain to me what is meant with the interval $[1,2]\times[0,\pi]$?

I did this, but I don't know how to continue.

$$\int_{[1,2]\times[0,\pi]} \log(\sqrt{x})\sin(2y)~d(x,y) = \dfrac{\sin\left(2y\right)\ln\left(x\right)}{2} = \frac{\sin(2y)}{2} \int{\ln(x)}~dx$$

Here's what the function looks like, it looks nice imo.

Answer 1

Hint: the integral means $\int_1^2 \int_0^{\pi}$. Further, $$\int_1^2\int_0^\pi f(x)g(y)\,dy\,dx = \left(\int_1^2 f(x)\,dx\right)\left(\int_0^\pi g(y)\,dy\right).$$