The image depicts a 3D solid created by revolving the region under the curve $y = \sqrt{x}$ from $x = 0$ to $x = 5$ around the x-axis. To find the volume of the solid, we can use the method of disks.

Volume using the Disk Method:

In this method, the volume of a solid of revolution is computed by integrating the area of infinitesimally thin circular disks along the axis of rotation. The formula for the volume is given by:

$V = \int_{a}^{b} \pi [f(x)]^2 \, dx$

Where:

• $f(x) = \sqrt{x}$ (the given curve),
• $a = 0$ and $b = 5$ are the limits of integration (bounds).

Step-by-step process:

1. Curve equation: $y = \sqrt{x}$.
2. Disk radius: Since the solid is revolved around the x-axis, the radius of each disk is $y = \sqrt{x}$.
3. Disk area: The area of a disk is $\pi \times (\text{radius})^2 = \pi (\sqrt{x})^2 = \pi x$.
4. Volume integral: The total volume is given by the integral of the disk areas from $x = 0$ to $x = 5$:

$V = \int_{0}^{5} \pi x \, dx$

Solving the integral:

$V = \pi \int_{0}^{5} x \, dx = \pi \left[ \frac{x^2}{2} \right]_0^5 = \pi \left( \frac{5^2}{2} - \frac{0^2}{2} \right)$ $V = \pi \left( \frac{25}{2} \right) = \frac{25\pi}{2}$

Final Volume:

$V = \frac{25\pi}{2} \, \text{cubic units}$

Thus, the volume of the solid generated by revolving the curve $y = \sqrt{x}$ from $x = 0$ to $x = 5$ around the x-axis is $\frac{25\pi}{2}$ cubic units.

Do you want further details or have any other questions?

Here are 5 relative questions for deeper understanding:

1. How does the disk method compare to the shell method for calculating volume?
2. What would change in the formula if we revolved the curve around the y-axis instead of the x-axis?
3. Can you apply the washer method to this problem if there were another curve?
4. How would the volume formula change if the curve was $y = x^2$ instead?
5. What are the physical interpretations of this volume in real-world applications?

Tip: Always check the bounds carefully when applying volume formulas to ensure you're integrating over the correct region.