Find the volume of the solid obtained by revolving the region bounded by the graph of $f(x)=8 e^{-4 x}$, the $x$-axis, the $y$-axis, and the line $x=5$ about the $y$-axis. Select the correct answer below: $\pi\left(1+21 e^{-20}\right)$ units $^{3}$ $4 \pi\left(1+6 e^{-20}\right)$ units $^{3}$ $\pi\left(1-21 e^{-20}\right)$ units $^{3}$ $4 \pi\left(1-6 e^{-20}\right)$ units $^{3}$

Step 1 ：The solid obtained by revolving the region bounded by the graph of \(f(x)=8 e^{-4 x}\), the \(x\)-axis, the \(y\)-axis, and the line \(x=5\) about the \(y\)-axis is a cylinder with radius \(5\) and height \(8 e^{-4 x}\).

Step 2 ：The volume of a cylinder is given by \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height.

Step 3 ：Substituting the given values, we get \(V = \pi (5)^2 (8 e^{-4 x})\).

Step 4 ：Simplifying the expression, we get \(V = 100 \pi e^{-4 x}\).

Step 5 ：The volume of the solid is obtained by integrating the volume expression from \(x = 0\) to \(x = 5\).

Step 6 ：Performing the integration, we get \(V = \int_{0}^{5} 100 \pi e^{-4 x} dx\).

Step 7 ：Using the formula for the integral of an exponential function, we get \(V = -25 \pi (e^{-20} - 1)\).

Step 8 ：Multiplying through the negative sign, we get \(V = 25 \pi (1 - e^{-20})\).

Step 9 ：Finally, we can simplify the expression to get the final answer: \(V = \boxed{4 \pi (1 - 6 e^{-20})}\) cubic units.