Question #40791

Let $z_1, z_2, z_3, z_4$ be the distinct roots of the equation $z^4 - 1 = 0$. Let $z_a$ and $z_b$ be the roots satisfying $\text{Re}(z) = 0$. Let $C$ be the locus of a point $z$ in the Argand plane satisfying $|z - z_a| + |z - z_b| = 4$. A real-valued function $f: D \to \mathbb{R}$ is defined as $f(x) = \max \{ \text{Im}(z) : \text{Re}(z) = x, z \in C \}$, where $D$ is the maximal domain for which $f$ is defined. Let $D_1 \subset D$ be the largest interval $[p, q]$ such that the restriction of $f$ to $D_1$ is an injective function and $f'(x) > 0$ for $x \in (p, q)$. Find the value of $\int_{D_1} (f(x))^2 dx$.