Answer
$\text{The volume is}$
\begin{align}
V = 9\pi
\end{align}
Work Step by Step
$\text{It is given that}$
\begin{align}
xy = 4 \ \ and \ \ x + y = 5 \Rrightarrow y = \frac{4}{x} \ \ and \ \ y = 5 -x
\end{align}
$\text{The intersections of these two functions are}$
\begin{align}
& \frac{4}{x} = 5 - x \Rrightarrow x^2 - 5x + 4 = 0 \Rrightarrow x = 4 \ \ and \ \ x = 1 \Rrightarrow \\ & \Rrightarrow y = 1 \ \ and \ \ y = 4
\end{align}
$\text{Thus, the volume is}$
\begin{align}
V = 2\pi\int_1^4 x \left(5-x - \frac{4}{x}\right) \ dx = 2\pi \left[\frac{5x^2}{2} - \frac{x^2}{2} - 4x \right]_1^4 = 2\pi \times \frac{9}{2} =9\pi
\end{align}
