To find the volume of the solid formed by rotating the region bounded by the curves $y = x^5$, $y = 0$, and $x = 1$ around the line $x = 2$, we can use the method of cylindrical shells.

Step 1: Understanding the problem

The region is bounded by:

• The curve $y = x^5$
• The line $y = 0$ (the x-axis)
• The vertical line $x = 1$

We are rotating this region around the vertical line $x = 2$.

Step 2: Formula for volume using cylindrical shells

The formula for the volume of a solid generated by rotating a region around a vertical line is given by:

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

For this problem:

• The radius of a shell is the distance from $x = 2$ to a point $x$, which is $|2 - x|$.
• The height of the shell is the function value, which is $y = x^5$.

Thus, the volume integral becomes:

$V = 2 \pi \int_{0}^{1} (2 - x) \cdot x^5 \, dx$

Step 3: Set up the integral

Now, let's compute the integral:

$V = 2 \pi \int_{0}^{1} (2x^5 - x^6) \, dx$

Step 4: Evaluate the integral

First, we integrate each term:

• $\int 2x^5 \, dx = \frac{2x^6}{6} = \frac{x^6}{3}$
• $\int x^6 \, dx = \frac{x^7}{7}$

Thus, the integral becomes:

$V = 2 \pi \left[ \frac{x^6}{3} - \frac{x^7}{7} \right]_{0}^{1}$

Step 5: Evaluate at the limits

Substitute $x = 1$ and $x = 0$ into the expression:

At $x = 1$:

$\frac{1^6}{3} - \frac{1^7}{7} = \frac{1}{3} - \frac{1}{7} = \frac{7}{21} - \frac{3}{21} = \frac{4}{21}$

At $x = 0$, both terms vanish.

So, the volume is:

$V = 2 \pi \left( \frac{4}{21} \right)$

Thus, the volume is:

$V = \frac{8 \pi}{21}$

Final Answer:

The volume of the solid is $\frac{8 \pi}{21}$ cubic units.

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Would you like further clarification or a breakdown of the steps?

Related questions:

1. How would the volume change if we rotated around a different vertical line (e.g., $x = 1$)?
2. What would the volume be if we used the disk method instead of cylindrical shells?
3. How do you compute the volume of solids of revolution with other types of functions (like trigonometric functions)?
4. What is the difference between rotating around a vertical line versus a horizontal line?
5. How would the volume change if the limits of integration changed (e.g., from $x = 0$ to $x = 2$)?

Tip: When setting up cylindrical shells, always carefully check the radius of each shell relative to the axis of rotation, and ensure the limits of integration match the region being rotated.