I am having some trouble understanding how to apply Fubini and/or Tonelli Theorems to determine whether a Lebesgue integral over $\mathbb{R}^2_+$ exists and if it is finite.
If someone could help me by showing the explicit steps for the examples below I would be grateful. I have a long list of exercises I have found online (this is self-study) and instead of posting a bunch of examples here I thought a few simple ones would help me learn how to do these problems going forward.
The examples in hand are: for each of the functions, use Fubini or Tonelli to show the existence/finiteness of the function's Lebesgue integral over $\mathbb{R}^2_+$.
$f_1(x,y)=\frac{\sin xy}{xy}$
$f_2(x,y)=e^{-(1+x^2)y}$
Many thanks in advance!