To find the volume of the solid formed by revolving the region bounded by the curves $y = x^2 + 1$, $x = 0$, $x = 1$, and $y = 0$ about the line $x = 1$, we can use the method of cylindrical shells. This method is particularly useful when revolving a region around a vertical line that is not one of the axes.

Step-by-Step Solution:

1. Region Boundaries:

   • $y = x^2 + 1$ (upper boundary)
   • $y = 0$ (lower boundary)
   • $x = 0$ (left boundary)
   • $x = 1$ (right boundary)

2. Cylindrical Shells Method: When rotating about the vertical line $x = 1$, the formula for the volume of a cylindrical shell is:

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

   • Radius: The distance from the line of revolution $x = 1$ to a point $x$ is $|1 - x| = 1 - x$.
   • Height: The height of the shell is the function $y = x^2 + 1$.

   So, the volume integral becomes: $V = 2\pi \int_{0}^{1} (1 - x)(x^2 + 1) \, dx$

3. Expand the integrand: $(1 - x)(x^2 + 1) = (1 \cdot x^2) + (1 \cdot 1) - (x \cdot x^2) - (x \cdot 1)$ Simplifying: $= x^2 + 1 - x^3 - x$

   Thus, the integral becomes: $V = 2\pi \int_{0}^{1} (x^2 + 1 - x^3 - x) \, dx$

4. Integrate each term: $\int (x^2 + 1 - x^3 - x) \, dx = \left[\frac{x^3}{3} + x - \frac{x^4}{4} - \frac{x^2}{2}\right]_{0}^{1}$

5. Evaluate the integral at the bounds $0$ and $1$:

   • At $x = 1$: $\frac{1^3}{3} + 1 - \frac{1^4}{4} - \frac{1^2}{2} = \frac{1}{3} + 1 - \frac{1}{4} - \frac{1}{2}$ Combine the fractions: $= \frac{1}{3} + 1 - \frac{1}{4} - \frac{1}{2} = \frac{4}{12} + \frac{12}{12} - \frac{3}{12} - \frac{6}{12}$ Simplifying: $= \frac{4 + 12 - 3 - 6}{12} = \frac{7}{12}$

   • At $x = 0$, all terms are 0.

6. Multiply by $2\pi$: $V = 2\pi \times \frac{7}{12} = \frac{7\pi}{6}$

Final Answer:

The volume of the solid is $\frac{7\pi}{6}$.

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Would you like more details or have any further questions? Here are some related problems for you to explore:

1. Find the volume if the region is revolved around the line $x = 2$.
2. What is the volume when the same region is revolved around the y-axis?
3. Find the centroid of the region before revolving.
4. Use the disk method to compute the volume revolving around the x-axis.
5. How does the volume change if the upper boundary is $y = 2x^2 + 1$?

Tip: The method of cylindrical shells is generally easier for problems involving revolutions around vertical lines.