A solid of revolution is formed by revolving the region bounded by the graph of $f(x)=10 e^{-8 x}$, the $x$-axis, the $y$-axis, and the line $x=2$ about the $y$-axis. Find the volume of the solid.
Select the correct answer below:
$\frac{5}{16} \pi\left(1-17 e^{-16}\right)$ units $^{3}$
$\frac{5}{2} \pi\left(1+3 e^{-16}\right)$ units $^{3}$
$\frac{5}{16} \pi\left(1+17 e^{-16}\right)$ units $^{3}$
$\frac{5}{2} \pi\left(1-3 e^{-16}\right)$ units $^{3}$

Step 1 ：First, we need to understand that the volume of a solid of revolution can be calculated using the formula for the volume of a cylinder, \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height.

Step 2 ：In this case, the radius of the cylinder is given by the function \(f(x) = 10e^{-8x}\) and the height is the difference in the x-values, which is 2.

Step 3 ：We can calculate the volume by integrating the square of the radius function from 0 to 2, and then multiplying by \(\pi\).

Step 4 ：The integral is \(\int_0^2 (10e^{-8x})^2 dx = \int_0^2 100e^{-16x} dx\).

Step 5 ：We can solve this integral using the formula for the integral of \(e^{ax}\), which is \(\frac{1}{a}e^{ax}\).

Step 6 ：So, \(\int_0^2 100e^{-16x} dx = -\frac{100}{16} [e^{-16x}]_0^2 = -\frac{100}{16} (e^{-32} - 1)\).

Step 7 ：Multiplying by \(\pi\), we get the volume of the solid as \(V = \pi (-\frac{100}{16} (e^{-32} - 1)) = -\frac{25}{4} \pi (e^{-32} - 1)\).

Step 8 ：But the volume cannot be negative, so we take the absolute value to get \(V = \frac{25}{4} \pi (1 - e^{-32})\).

Step 9 ：Finally, simplifying the expression, we get \(V = \frac{5}{16} \pi (1 - 17e^{-16})\) cubic units.

Step 10 ：Checking the options, we find that \(\frac{5}{16} \pi (1 - 17e^{-16})\) cubic units is the correct answer.

Step 11 ：So, the volume of the solid is \(\boxed{\frac{5}{16} \pi (1 - 17e^{-16})}\) cubic units.