The volume of the solid obtained by rotating the region enclosed by \[ y=1 / x^{3}, \quad y=0, \quad x=2, \quad x=4 \] about the line $x=-4$ can be computed using the method of cylindrical shells via an integral \[ V=\int_{a}^{b} \square ? \] with limits of integration $a=$ and $b=$

Step 1 ：The volume of a solid obtained by rotating a region about a line can be computed using the method of cylindrical shells. The formula for the volume is given by: \[V = 2\pi \int_{a}^{b} r(x)h(x) dx\] where \(r(x)\) is the radius of the cylindrical shell at \(x\) and \(h(x)\) is the height of the cylindrical shell at \(x\).

Step 2 ：In this case, the region is bounded by \(y=1/x^3\), \(y=0\), \(x=2\), and \(x=4\), and is rotated about the line \(x=-4\).

Step 3 ：The radius of the cylindrical shell at \(x\) is the distance from \(x\) to the line \(x=-4\), which is \(x - (-4) = x + 4\).

Step 4 ：The height of the cylindrical shell at \(x\) is the distance from \(y=0\) to \(y=1/x^3\), which is \(1/x^3 - 0 = 1/x^3\).

Step 5 ：So, we have \(r(x) = x + 4\) and \(h(x) = 1/x^3\).

Step 6 ：The limits of integration are the values of \(x\) that bound the region, which are \(x=2\) and \(x=4\).

Step 7 ：So, the integral that gives the volume of the solid is: \[V = 2\pi \int_{2}^{4} (x + 4)(1/x^3) dx\]

Step 8 ：Now, we can compute this integral to find the volume. The final answer is \(\boxed{\frac{5\pi}{4}}\).