To find the volume of the solid generated when the given region is revolved around the $y$-axis, we can use the method of cylindrical shells or the disk/washer method. However, the shell method is more convenient here as the region is bounded in terms of $y$.

Problem setup:

1. Equations of boundaries:

   • $y = 10 - x$ (slant line),
   • $y = 10$ (horizontal line),
   • $y = x$ (diagonal line).

2. Intersection points:

   • The lines $y = 10 - x$ and $y = x$ intersect at $x = 5$ (substitute $x = 10 - x$ → $x = 5$).
   • The region is bounded by $x = 0$ to $x = 5$ along the $y$-axis.

Shell method formulation:

When using the shell method, the formula for the volume is:

$V = 2\pi \int_{x=a}^{x=b} \text{(radius)} \cdot \text{(height)} \, dx$

• Radius: The distance of the shell from the $y$-axis is $x$.
• Height: The height is determined by $y_{\text{top}} - y_{\text{bottom}}$, which is $(10 - x) - x = 10 - 2x$.
• Bounds: $x$ ranges from 0 to 5.

Thus, the volume is: $V = 2\pi \int_{0}^{5} x(10 - 2x) \, dx$

Solve the integral:

1. Expand the integrand: $x(10 - 2x) = 10x - 2x^2$

2. Set up the integral: $V = 2\pi \int_{0}^{5} (10x - 2x^2) \, dx$

3. Integrate term by term: $\int (10x) \, dx = 5x^2, \quad \int (2x^2) \, dx = \frac{2x^3}{3}$

   Therefore: $V = 2\pi \left[ 5x^2 - \frac{2x^3}{3} \right]_{0}^{5}$

4. Evaluate at the bounds: At $x = 5$: $5(5^2) - \frac{2(5^3)}{3} = 125 - \frac{250}{3} = \frac{375}{3} - \frac{250}{3} = \frac{125}{3}$

   At $x = 0$: $5(0^2) - \frac{2(0^3)}{3} = 0$

   So: $V = 2\pi \left( \frac{125}{3} - 0 \right) = \frac{250\pi}{3}$

Final Answer:

$\boxed{\frac{250\pi}{3}}$

Let me know if you'd like a detailed explanation of a different method or further clarifications.

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Related Questions:

1. How does the washer method differ from the shell method?
2. How would the result change if the region were revolved around the $x$-axis instead of the $y$-axis?
3. Can you visualize the solid formed by this revolution?
4. What if we included additional constraints, like $y > 5$?
5. How can numerical integration confirm the solution?

Tip:

When choosing a method for volume, prioritize clarity and alignment of the boundaries (vertical or horizontal) to simplify calculations.