  Let $\Omega$ be a convex domain in the complex plane ${\mathbb C}$ with
$\Omega \not= {\mathbb C}$, and $P$ be a conformal map of the unit disk
${\mathbb D}$ onto $\Omega$. Let ${\mathcal F}_\Omega$ be the class of analytic
functions $g$ in ${\mathbb D}$ with $g({\mathbb D}) \subset \Omega$, and
$H_1^\infty ({\mathbb D})$ be the closed unit ball of the Banach space
$H^\infty ({\mathbb D})$ of bounded analytic functions $\omega$ in ${\mathbb
D}$, with norm $\| \omega \|_\infty = \sup_{z \in {\mathbb D}} |\omega (z)|$.
Let ${\mathcal C}(n) = \{ (c_0,c_1 , \ldots , c_n ) \in {\mathbb C}^{n+1}:
\text{there exists} \; \omega \in H_1^\infty ({\mathbb D}) \; \text{satisfying}
\; \omega (z) = c_0+c_1z + \cdots + c_n z^n + \cdots$ for ${z\in \mathbb D}\}$.
For each fixed $z_0 \in {\mathbb D}$, $j=-1,0,1,2, \ldots$ and $c = (c_0, c_1 ,
\ldots , c_n) \in {\mathcal C}(n)$, we use the Schur algorithm to determine the
region of variability $V_\Omega^j (z_0, c ) = \{ \int_0^{z_0}
z^{j}(g(z)-g(0))\, d z : g \in {\mathcal F}_\Omega \; \text{with} \; (P^{-1}
\circ g) (z) = c_0 +c_1z + \cdots + c_n z^n + \cdots \}$. We also show that for
$z_0 \in {\mathbb D} \backslash \{ 0 \}$ and $c \in \textrm{Int} \, {\mathcal
C}(n) $, $V_\Omega^j (z_0, c )$ is a convex closed Jordan domain, which we
determine by giving a parametric representation of the boundary curve $\partial
V_\Omega^j (z_0, c )$.
