Solution
= Solution
Use <cylindrical coordinates> $x=r\cos\theta$, $y=r\sin\theta$. The upper plane gives $z=4-r(\cos\theta+\sin\theta)>0$ throughout the quarter disk, and the <Jacobian determinant> contributes $r$. Therefore
$$
\begin{aligned}
V&=\int_0^{\pi/2}\int_0^1[4-r(\cos\theta+\sin\theta)]r\,dr\,d\theta\\
&=\int_0^{\pi/2}\left[2-\frac{\cos\theta+\sin\theta}{3}\right]d\theta,
\end{aligned}
$$
so
$$
\boxed{V=\pi-\frac23.}
$$