To find the volume of the solid of revolution generated by revolving the region bounded by $y = 4x - x^2$ and the x-axis around the x-axis using the shell method, we follow these steps:

1. Define the bounds and the function:

The curve $y = 4x - x^2$ represents the shape of the region. The region is bounded by the x-axis ($y = 0$) and the curve. We need to determine where the curve intersects the x-axis, which occurs when $y = 0$: $4x - x^2 = 0$ Factor the equation: $x(4 - x) = 0$ So, $x = 0$ or $x = 4$. The region of interest is between $x = 0$ and $x = 4$.

2. Set up the shell method formula:

The shell method for solids of revolution about the x-axis uses the formula: $V = 2\pi \int_{a}^{b} \text{(radius)} \times \text{(height)} \, dx$ For this problem:

• The radius of a shell at a point $x$ is simply $y = 4x - x^2$.
• The height is the distance from the x-axis, which is also $y = 4x - x^2$.

Thus, the volume integral becomes: $V = 2\pi \int_{0}^{4} x \cdot (4x - x^2) \, dx$

3. Simplify the integrand:

Distribute $x$ inside the parentheses: $V = 2\pi \int_{0}^{4} (4x^2 - x^3) \, dx$

4. Integrate:

Now, integrate the function: $\int (4x^2 - x^3) \, dx = \frac{4x^3}{3} - \frac{x^4}{4}$

5. Evaluate the definite integral:

Evaluate the expression from $0$ to $4$: $V = 2\pi \left[ \left( \frac{4x^3}{3} - \frac{x^4}{4} \right) \Big|_0^4 \right]$

At $x = 4$: $\frac{4(4)^3}{3} - \frac{(4)^4}{4} = \frac{4 \times 64}{3} - \frac{256}{4} = \frac{256}{3} - 64$ $= \frac{256}{3} - \frac{192}{3} = \frac{64}{3}$

At $x = 0$, both terms are 0.

So, the volume is: $V = 2\pi \times \frac{64}{3} = \frac{128\pi}{3}$

Final Answer:

The volume of the solid of revolution is $\frac{128\pi}{3}$ cubic units.

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Would you like more details on any part of the solution? Here are some related questions you might find helpful:

1. How would the result change if the region were revolved about the y-axis instead?
2. What is the physical interpretation of the shell method in this context?
3. How do we determine the volume of a solid of revolution using the disk method?
4. What are some common errors to avoid when using the shell method?
5. How would you solve for the volume of a region defined by a different function?

Tip: When using the shell method, always remember that the radius of the shell is the distance from the axis of revolution (in this case, the x-axis) to the function, while the height is the function itself.