To find the total volume $V$ of the solid obtained by rotating the region under the curve $y = x^{1/5}$ around the x-axis from $x = 0$ to $x = 2$, we follow these steps:

Step 1: Area of the circular cross-section

The area of a circular cross-section at a generic point $x$ is given by: $A(x) = \pi \left( y(x) \right)^2 = \pi \left( x^{1/5} \right)^2 = \pi x^{2/5}.$

Step 2: Volume of an approximating disk

The volume of an approximating disk at a point $x$ with a small thickness $\Delta x$ is: $\text{Volume of disk} = A(x) \Delta x = \pi x^{2/5} \Delta x.$

Step 3: Total volume of the solid

To find the total volume, sum up the volumes of the disks as $\Delta x$ approaches 0. This is done by integrating the area function $A(x)$ from $x = 0$ to $x = 2$: $V = \int_0^2 A(x) \, dx = \int_0^2 \pi x^{2/5} \, dx.$ We can factor out the constant $\pi$: $V = \pi \int_0^2 x^{2/5} \, dx.$

Step 4: Compute the integral

To compute $\int_0^2 x^{2/5} \, dx$, use the power rule of integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1}, \quad \text{where } n = \frac{2}{5}.$ So, $\int_0^2 x^{2/5} \, dx = \left[ \frac{x^{7/5}}{7/5} \right]_0^2 = \frac{5}{7} \left( 2^{7/5} - 0^{7/5} \right).$ Since $0^{7/5} = 0$, this simplifies to: $\int_0^2 x^{2/5} \, dx = \frac{5}{7} \cdot 2^{7/5}.$

Step 5: Final volume expression

Now, multiplying by $\pi$: $V = \pi \cdot \frac{5}{7} \cdot 2^{7/5}.$

Thus, the total volume of the solid is: $V = \frac{5\pi}{7} \cdot 2^{7/5}.$

Would you like a numerical evaluation of this result or more details on any step?

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Here are 5 related questions to expand on this concept:

1. How would the volume change if the curve was rotated about the y-axis instead?
2. What is the volume if the curve is $y = \sqrt{x}$ from $x = 0$ to $x = 2$?
3. How does the integral setup change if we rotate around a horizontal line $y = c$?
4. How can we find the surface area of the same solid instead of the volume?
5. What happens if we rotate the region under the curve about the line $x = 3$?

Tip: For solids of revolution, always check if the cross-section is perpendicular to the axis of rotation to correctly apply the disk or washer method.