In coordinate space, consider the circle of radius $1$ centered at the origin in the $xy$-plane. Let $S$ be the cone (including its interior) with this circle as its base and with vertex at the point $(0,\,0,\,2)$. Also, let $A(1,\,0,\,2)$.
[(1)] When point $P$ moves over the base of $S$, let $T$ be the region swept out by the line segment $AP$. Illustrate in the same plane the cross-section of $S$ by the plane $z=1$ and the cross-section of $T$ by the plane $z=1$.
[(2)] When point $P$ moves throughout $S$, find the volume of the region swept out by the line segment $AP$.
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In coordinate space, consider the circle of radius $1$ centered at the origin in the $xy$-plane. Let $S$ be the cone (including its interior) with this circle as its base and with vertex at the point $(0,\,0,\,2)$. Also, let $A(1,\,0,\,2)$.
\begin{enumerate}
\item[(1)] When point $P$ moves over the base of $S$, let $T$ be the region swept out by the line segment $AP$. Illustrate in the same plane the cross-section of $S$ by the plane $z=1$ and the cross-section of $T$ by the plane $z=1$.
\item[(2)] When point $P$ moves throughout $S$, find the volume of the region swept out by the line segment $AP$.
\end{enumerate}
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