Suppose that D is the unit disk in the xy-plane, which is contained in the unit square S = [0, 1] × [0, 1]. Let ƒ and g be the functions f1 if (x, y) E D if (x, y) E S – D' if (x, y) E S – D if (x, y) E D -1 f (x, y) : g(x, y) = and define Sf(x, y)0(r9) if (x, y) € D 9(x, y){(-3) if (x, y) E S – D F(x, y) Compute the value of /| F(x, y) dA.
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2. Please give as much detail as possible
![Suppose that \( D \) is the unit disk in the \( xy \)-plane, which is contained in the unit square \( S = [0,1] \times [0,1] \). Let \( f \) and \( g \) be the functions
\[
f(x, y) =
\begin{cases}
1 & \text{if } (x, y) \in D \\
0 & \text{if } (x, y) \in S - D
\end{cases}
, \quad
g(x, y) =
\begin{cases}
-1 & \text{if } (x, y) \in S - D \\
0 & \text{if } (x, y) \in D
\end{cases}
\]
and define
\[
F(x, y) =
\begin{cases}
f(x, y)^{g(x, y)} & \text{if } (x, y) \in D \\
g(x, y)^{f(x, y)} & \text{if } (x, y) \in S - D
\end{cases}
\]
Compute the value of
\[
\iint_S F(x, y) \, dA.
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