Solution
= Solution
Put
$$
h(x)=u_+(x)-u_-(x).
$$
Vertical slicing and Fubini's theorem give
$$
\mathcal A(\Omega)=\int_{x_0}^{x_1}h(x)\,dx.
$$
The vertical slice of $S_L(\Omega)$ is $(-h(x)/2,h(x)/2)$ and has the same length $h(x)$. Therefore
$$
\boxed{\mathcal A(S_L(\Omega))
=\int_{x_0}^{x_1}h(x)\,dx
=\mathcal A(\Omega).}
$$