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To solve this problem, we need to find the volume of the solid generated by revolving the region bounded by the curve \( y = 2x^2 \), the x-axis (\( y = 0 \)), and the vertical line \( x = 4 \) around the x-axis.

### Step-by-step approach:

1. **Set up the integral using the disk method**:
   The volume of the solid generated by revolving a region around the x-axis is given by:
   \[
   V = \pi \int_{a}^{b} [f(x)]^2 \, dx
   \]
   where \( f(x) \) is the function describing the boundary of the region being revolved, and the limits of integration \( a \) and \( b \) correspond to the x-values that bound the region.

2. **Identify the function and bounds**:
   Here, \( f(x) = 2x^2 \), and we are revolving the region from \( x = 0 \) to \( x = 4 \).

3. **Substitute the function and bounds into the integral**:
   \[
   V = \pi \int_0^4 (2x^2)^2 \, dx
   \]
   Simplify the function inside the integral:
   \[
   V = \pi \int_0^4 4x^4 \, dx
   \]

4. **Evaluate the integral**:
   \[
   V = \pi \left[ \frac{4x^5}{5} \right]_0^4
   \]
   Now, compute the definite integral:
   \[
   V = \pi \left( \frac{4(4)^5}{5} - \frac{4(0)^5}{5} \right)
   \]
   \[
   V = \pi \left( \frac{4(1024)}{5} \right)
   \]
   \[
   V = \pi \left( \frac{4096}{5} \right)
   \]
   \[
   V = \frac{4096\pi}{5}
   \]

Thus, the volume of the solid is \( \frac{4096\pi}{5} \) cubic units.

Would you like further details on any step? 

Here are five questions to further explore the topic:

1. How does the disk method differ from the shell method for finding volumes of revolution?
2. What would change in the volume calculation if the region were revolved around a different axis?
3. Can you use this method for functions that are not continuous?
4. What happens if the function being revolved has negative values in the bounded region?
5. How would the process differ if the curve was a linear function instead of a quadratic?

**Tip**: Always check whether your function crosses the axis you're revolving around, as this can affect the integral limits or method.

Find the volume of the solid generated by revolving the region bounded by y = 2x^2, y = 0, and x = 4 about the x-axis.

Learn how to calculate the volume of a solid generated by revolving the region bounded by y = 2x^2, y = 0, and x = 4 about the x-axis using the disk method. This step-by-step solution uses calculus and the definite integral to find the volume, which results in 4096π/5 cubic units.

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