We are tasked with finding the volume of the solid formed by rotating the region bounded by the curve $y = -x^2 + 10x - 16$ and the line $y = 0$ (the x-axis) about the x-axis.

Step 1: Find the bounds

We first need to determine where the curve intersects the x-axis. This is done by solving for $x$ when $y = 0$:

$0 = -x^2 + 10x - 16$

This is a quadratic equation. We can solve it using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $-x^2 + 10x - 16 = 0$, we have $a = -1$, $b = 10$, and $c = -16$. Plugging these values into the quadratic formula:

$x = \frac{-10 \pm \sqrt{10^2 - 4(-1)(-16)}}{2(-1)}$ $x = \frac{-10 \pm \sqrt{100 - 64}}{-2}$ $x = \frac{-10 \pm \sqrt{36}}{-2}$ $x = \frac{-10 \pm 6}{-2}$

This gives two solutions:

$x_1 = \frac{-10 + 6}{-2} = 2, \quad x_2 = \frac{-10 - 6}{-2} = 8$

So, the curve intersects the x-axis at $x = 2$ and $x = 8$.

Step 2: Set up the volume integral (Disk method)

The volume of the solid formed by rotating the region about the x-axis can be found using the disk method. The formula for the volume is:

$V = \pi \int_{a}^{b} [f(x)]^2 \, dx$

In our case, $f(x) = -x^2 + 10x - 16$, and the limits of integration are from $x = 2$ to $x = 8$. So the volume is:

$V = \pi \int_{2}^{8} \left[ -x^2 + 10x - 16 \right]^2 \, dx$

Step 3: Simplify the integrand

We now expand $(-x^2 + 10x - 16)^2$:

$\left(-x^2 + 10x - 16\right)^2 = (x^4 - 20x^3 + 132x^2 - 320x + 256)$

Step 4: Perform the integration

We integrate each term:

$\int_{2}^{8} \left( x^4 - 20x^3 + 132x^2 - 320x + 256 \right) \, dx$

Each term is integrated as follows:

$\int x^4 \, dx = \frac{x^5}{5}, \quad \int x^3 \, dx = \frac{x^4}{4}, \quad \int x^2 \, dx = \frac{x^3}{3}, \quad \int x \, dx = \frac{x^2}{2}, \quad \int 1 \, dx = x$

Thus, the integral becomes:

$V = \pi \left[ \frac{x^5}{5} - 5x^4 + 44x^3 - 160x^2 + 256x \right]_{2}^{8}$

Step 5: Compute the definite integral

We now evaluate this expression at $x = 8$ and $x = 2$ and subtract the results:

For $x = 8$:

[ \frac{8^5}{5} - 5(