https://file.aiquickdraw.com/mathbot/file/1729308518775pyi2zuyw.jpg

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.

Find the volume of the solid formed by revolving the curve y = √x from x = 0 to x = 5 around the x-axis.

Learn how to calculate the volume of the solid formed by revolving the curve y = √x around the x-axis from x = 0 to x = 5. This detailed solution involves the disk method and is suitable for students in Calculus (Grades 11-12).

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