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Let $f(\vec{x})$ be a bounded function on a nonempty and bounded domain, $S\subseteq\mathbb{R}^n$. Define a function $g:I\to\mathbb{R}$, where $I$ is a box such that $S\subseteq I$, where
$$\begin{align*}
g(\vec{x}):=\begin{cases}f(\vec{x}) & \vec{x}\in S \\ 0 & \vec{x}\in I\backslash S\end{cases}
\end{align*}$$
We say that $f$ is Riemann integrable on S if $g$ is Riemann integrable on $I$, and we define
$$\begin{align*}
\int_{S}f(\vec{x})d\vec{x} = \int_I g(\vec{x}) d\vec{x}
\end{align*}$$
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Let $f(\vec{x})$ be a bounded function on a nonempty and bounded domain, $S\subseteq\mathbb{R}^n$. Define a function $g:I\to\mathbb{R}$, where $I$ is a box such that $S\subseteq I$, where
$$\begin{align*}
g(\vec{x}):=\begin{cases}f(\vec{x}) & \vec{x}\in S \\ 0 & \vec{x}\in I\backslash S\end{cases}
\end{align*}$$
We say that $f$ is Riemann integrable on S if $g$ is Riemann integrable on $I$, and we define
$$\begin{align*}
\int_{S}f(\vec{x})d\vec{x} = \int_I g(\vec{x}) d\vec{x}
\end{align*}$$
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