Find the volume of the solid generated by revolving the region enclosed by the triangle with vertices left parenthesis 3 comma 1 right parenthesis​, left parenthesis 3 comma 5 right parenthesis​, and left parenthesis 8 comma 5 right parenthesis about the​ y-axis.
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Part 1
The volume of the solid generated by revolving the region enclosed by the triangle with vertices left parenthesis 3 comma 1 right parenthesis​, left parenthesis 3 comma 5 right parenthesis and left parenthesis 8 comma 5 right parenthesis about the​ y-axis is 
  
enter your response here cubic units.

To find the volume of the solid generated by revolving the triangular region around the y-axis, we will use the **method of cylindrical shells**.

### Step 1: Analyze the Triangle

The triangle has the following vertices:
- \( (3, 1) \)
- \( (3, 5) \)
- \( (8, 5) \)

These points form a right triangle. The base of the triangle is from \( x = 3 \) to \( x = 8 \) at \( y = 5 \), and the height of the triangle is from \( y = 1 \) to \( y = 5 \) at \( x = 3 \).

### Step 2: Equation of the Line

We need the equation of the hypotenuse of the triangle, which connects the points \( (3, 1) \) and \( (8, 5) \).

The slope of the line is:
\[
m = \frac{5 - 1}{8 - 3} = \frac{4}{5}
\]
Thus, the equation of the line in point-slope form is:
\[
y - 1 = \frac{4}{5}(x - 3)
\]
Simplifying this equation:
\[
y = \frac{4}{5}(x - 3) + 1
\]
\[
y = \frac{4}{5}x - \frac{12}{5} + 1 = \frac{4}{5}x - \frac{7}{5}
\]
Thus, the equation of the line is:
\[
y = \frac{4}{5}x - \frac{7}{5}
\]

### Step 3: Volume by the Shell Method

Using the method of cylindrical shells, the volume is given by:
\[
V = 2\pi \int_{x_1}^{x_2} x \cdot h(x) \, dx
\]
where \( h(x) \) is the height of the shell, which is the difference between \( y = 5 \) and the equation of the line, and the limits of integration are from \( x = 3 \) to \( x = 8 \).

The height of the shell is:
\[
h(x) = 5 - \left( \frac{4}{5}x - \frac{7}{5} \right) = 5 - \frac{4}{5}x + \frac{7}{5} = \frac{25}{5} - \frac{4}{5}x + \frac{7}{5} = \frac{32}{5} - \frac{4}{5}x
\]

Thus, the volume integral is:
\[
V = 2\pi \int_3^8 x \left( \frac{32}{5} - \frac{4}{5}x \right) dx
\]
Expanding the integrand:
\[
V = 2\pi \int_3^8 \left( \frac{32}{5}x - \frac{4}{5}x^2 \right) dx
\]

### Step 4: Evaluate the Integral

Now, let's integrate each term:
\[
\int_3^8 \frac{32}{5}x \, dx = \frac{32}{5} \cdot \frac{x^2}{2} \Big|_3^8 = \frac{32}{10} \left( 8^2 - 3^2 \right) = \frac{16}{5}(64 - 9) = \frac{16}{5} \cdot 55 = 176
\]

\[
\int_3^8 \frac{4}{5}x^2 \, dx = \frac{4}{5} \cdot \frac{x^3}{3} \Big|_3^8 = \frac{4}{15} \left( 8^3 - 3^3 \right) = \frac{4}{15}(512 - 27) = \frac{4}{15} \cdot 485 = \frac{1940}{15} = 129.33
\]

### Step 5: Compute the Volume

Substitute the results into the volume formula:
\[
V = 2\pi \left( 176 - 129.33 \right) = 2\pi \cdot 46.67 = 293.54 \, \text{cubic units}
\]

Thus, the volume of the solid is approximately **293.54 cubic units**.

---

Would you like more details or explanations about any part of this process?

Here are 5 related questions for further exploration:
1. How would the result change if the region were revolved around the x-axis instead of the y-axis?
2. What is the significance of the cylindrical shells method in solving volume problems?
3. How can we derive the formula for the volume of solids of revolution using the disk/washer method?
4. What are other shapes that can be formed by revolving different polygons around an axis?
5. How would you compute the surface area of the solid generated by this revolution?

**Tip:** When revolving regions around an axis, carefully examine the boundaries and equations of the shapes involved to set up integrals correctly.

Find the volume of the solid generated by revolving the region enclosed by the triangle with vertices (3, 1), (3, 5), and (8, 5) about the y-axis.

Discover how to calculate the volume of a solid generated by revolving a triangle with vertices (3, 1), (3, 5), and (8, 5) around the y-axis using the cylindrical shells method. This detailed solution includes key calculus concepts and integration steps, ideal for high school and college students.

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