The set is $M=\{(x,y)\in\mathbb{R}^2:|x|+|y|\leq 1\}$.
Question: How do you calculate the area of $M$? More specific, how do you find the bounds of integration?
Attempt: I tried to solve the inequality, yielding, $$ \begin{align*} (x,y)>0&:x\leq -y+1,y\leq 1&(x,y)<0&:-x\leq y+1\Leftrightarrow x\geq -y-1,y\geq -1\\ \end{align*} $$ and then the integral $$ \begin{align*} \mathcal{J}&=\int_0^1\int_0^{-y+1}\text{d}x\text{d}y-\int_{-1}^0\int_{-y-1}^0\text{d}x\text{d}y=0 .\end{align*} $$ The solution, however, is $\mathcal{J}'=2$. I presume, the bounds of integration are wrong, thus the solution to the inequality is not correct.
Edit: Corrected a typo in the inequality, but did not change integration bounds (see comments).
$A=\iint_M \mathrm dx \mathrm dy = 2\int_{-1}^0\mathrm dx \int_0^{1+x} \mathrm dy + 2 \int_0^1 \mathrm dx \int_0^{1-x} \mathrm dy = \dots = 2\dfrac{1}{2} + 2\dfrac{1}{2} = \boxed{2}$.
Figure: