Let $f(x_1,x_2)=\frac{x_1}{x_2^{1/3}} \times \chi_{\{ [0,1] \times[-1,1]\}}$.
Part A: Evaluate $\int_{\mathbb{R}^2} f(x_1,x_2) ~ \mathrm{d}x_1~ \mathrm{d}x_2$
$\int_{\mathbb{R}^2} f(x_1,x_2)~ \mathrm{d}x_1~ \mathrm{d}x_2=\int_{-1}^1 \int_0^1 x_1x_2^{-1/3} ~ \mathrm{d}x_1~ \mathrm{d}x_2=\int_{-1}^1 \frac{x_1^2}{2} \times x_2^{-1/3} \Big]_0^1\mathrm{d}x_2=\int_{-1}^1 \frac{x_2^{-1/3}}{2} ~ \mathrm{d}x_2=\frac{1}{2}\frac{x_2^{2/3}}{3/2}\Big]_{-1}^1=\frac{1}{3}(1-1)=0$
Part B: For which values of $p \geq 1$ is $f \in L^p(\mathbb{R^2})$?
How would I do this?
The technique is basically the same as in Part A, but you need to be careful about using Fubini's theorem. Fix some set $U = [\epsilon, 1] \times[-1, 1]$ for $\epsilon > 0$. Then $$C_1 \int_{[-1, 1]} |x_2|^{-p/3}\,dx_2 \leq \int_U |f|^p\, \leq C_2 \int_{[-1, 1]} |x_2|^{-p/3}\, dx_2,$$ where $C_1 = \min_U |x|^p > 0$ and $C_2 =\max_U |x|^p > 0$. That should give you good enough bounds on $\|f\|_p$ to complete part B.