We are asked to find the volume of the solid generated by revolving a triangle about different axes. The vertices of the triangle are $(2, 2)$, $(2, 8)$, and $(6, 8)$. First, let's handle part a, revolving around the $x$-axis.

Part a: Revolving the Triangle About the $x$-Axis

We can use the method of disks to calculate the volume when revolving around the $x$-axis. The general formula for the volume of revolution using the disk method is:

$V = \pi \int_a^b [f(x)]^2 \, dx$

Here, the height of the triangle above the $x$-axis is given by the vertical distance of the triangle from the $x$-axis (the $y$-values). Since we are rotating the triangle, we must express the edges of the triangle as functions of $x$.

1. The vertical line from $(2, 8)$ to $(2, 2)$ is simply a constant line at $x = 2$.
2. The line from $(2, 8)$ to $(6, 8)$ is a horizontal line at $y = 8$.

We can break the integral into two parts to capture the geometry of the triangle. First, we need the equation of the slanted line that connects $(2, 2)$ and $(6, 8)$. The slope $m$ of the line is:

$m = \frac{8 - 2}{6 - 2} = \frac{6}{4} = 1.5$

Thus, the equation of the line is:

$y - 2 = 1.5(x - 2) \quad \Rightarrow \quad y = 1.5x - 1$

Now, we integrate from $x = 2$ to $x = 6$ using the disk method with the function $y(x)$.

Volume Calculation

Using the disk method, the formula for the volume becomes:

$V = \pi \int_2^6 [y(x)]^2 \, dx$

Substitute the expression for $y(x)$:

$V = \pi \int_2^6 (1.5x - 1)^2 \, dx$

Let's calculate this integral.The volume of the solid generated by revolving the triangle around the $x$-axis is $112\pi$ cubic units.

Do you want more details on the solution or have any questions? Here are five related questions for further exploration:

1. How does the method of cylindrical shells compare to the disk method for revolving around the $x$-axis?
2. What is the volume generated if the triangle is revolved around the line $y = 8$ instead?
3. How does changing the limits of integration affect the volume?
4. Can you derive the equation of a curve if the shape was not a straight-edged triangle but a curved region?
5. How would the result differ if the triangle had different vertex coordinates?

Tip: When solving volume problems involving rotation, visualize the 3D shape to understand which method (disk, washer, or shell) is best for the given axis of rotation.