The volume of the solid of revolution is given by the formula:

\(V = \pi \int (f(x)^2 - g(x)^2) \, dx\)

where \(f(x) = 5 - x\) and \(g(x) = \frac{4}{x}\).

Calculate the volume by integrating from \(x = 1\) to \(x = 4\):

\(V = \pi \left( \int_1^4 (5-x)^2 \, dx - \int_1^4 \frac{16}{x^2} \, dx \right)\)

First, integrate \((5-x)^2\):

\(\int (5-x)^2 \, dx = \frac{(5-x)^3}{3}\)

Evaluate from 1 to 4:

\(\left[ \frac{(5-x)^3}{3} \right]_1^4 = \frac{(5-4)^3}{3} - \frac{(5-1)^3}{3} = \frac{1}{3} - \frac{64}{3} = -\frac{63}{3} = -21\)

Next, integrate \(\frac{16}{x^2}\):

\(\int \frac{16}{x^2} \, dx = -\frac{16}{x}\)

Evaluate from 1 to 4:

\(\left[ -\frac{16}{x} \right]_1^4 = -\frac{16}{4} + \frac{16}{1} = -4 + 16 = 12\)

Subtract the two results:

\(V = \pi (-21 - 12) = \pi (-33) = 33\pi\)

Thus, the volume is \(9\pi\) or approximately 28.3.