Find the volumes of the solids generated by revolving the triangle with vertices left parenthesis 2 comma 2 right parenthesis​, left parenthesis 2 comma 8 right parenthesis​, and left parenthesis 6 comma 8 right parenthesis about a. the​ x-axis, b. the​ y-axis, c. the line xequals7​, and d. the line yequals2.
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Part 1
a. The volume of the solid generated by revolving about the​ x-axis is 
  
enter your response here cubic units.

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.

Find the volumes of the solids generated by revolving the triangle with vertices (2, 2), (2, 8), and (6, 8) about a. the x-axis, b. the y-axis, c. the line x=7, and d. the line y=2.

This detailed solution shows how to calculate the volume of solids generated by revolving a triangle with vertices (2, 2), (2, 8), and (6, 8) about the x-axis, y-axis, and lines x=7 and y=2. The solution uses the disk method and includes key formulas and geometry concepts.

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