Find the volume of the solid obtained by rotating the region bounded by the curves x=y^(1/2), x=0, and y=3 about the x-axis.

To find the volume of the solid obtained by rotating the region bounded by the curves \( x = \sqrt{y} \), \( x = 0 \), and \( y = 3 \) about the x-axis, we use the method of disks or washers.

### Step 1: Understand the Problem
The region is bounded by:
- \( x = \sqrt{y} \)
- \( x = 0 \) (the y-axis)
- \( y = 3 \)

The rotation is about the x-axis.

### Step 2: Set Up the Integral
The volume of the solid of revolution is given by the integral:
\[
V = \pi \int_{a}^{b} [R(y)]^2 \, dy
\]
where \( R(y) \) is the radius of the disk at a given height \( y \), and the limits of integration \( a \) and \( b \) are the bounds on \( y \).

Here, the radius of the disk is \( R(y) = \sqrt{y} \), since the curve is given by \( x = \sqrt{y} \). The limits of integration are from \( y = 0 \) to \( y = 3 \).

### Step 3: Write the Integral
Substituting \( R(y) = \sqrt{y} \) into the formula for the volume, we get:
\[
V = \pi \int_{0}^{3} (\sqrt{y})^2 \, dy
\]

### Step 4: Simplify the Integral
Simplify the integrand:
\[
V = \pi \int_{0}^{3} y \, dy
\]

### Step 5: Evaluate the Integral
Now, integrate with respect to \( y \):
\[
V = \pi \left[ \frac{y^2}{2} \right]_{0}^{3}
\]
Substituting the limits:
\[
V = \pi \left[ \frac{3^2}{2} - \frac{0^2}{2} \right] = \pi \left[ \frac{9}{2} \right]
\]
\[
V = \frac{9\pi}{2}
\]

### Final Answer
The volume of the solid obtained by rotating the region about the x-axis is:
\[
\boxed{\frac{9\pi}{2}}
\]

Would you like any details or have further questions?

Here are 5 related questions to expand your understanding:
1. How would the volume change if the region was rotated about the y-axis instead?
2. What if the boundary curve was \( x = y^2 \) instead of \( x = \sqrt{y} \)?
3. How would you set up the integral if the region were rotated about the line \( y = 3 \) instead of the x-axis?
4. What is the geometric interpretation of the volume calculated using the disk method?
5. Can you derive the same volume using the method of cylindrical shells?

**Tip:** Always visualize the region and the axis of rotation to help set up the correct integral for volume calculations.

Discover how to find the volume of the solid obtained by rotating the region bounded by the curves x=y^(1/2), x=0, and y=3 about the x-axis. Step-by-step solution using the disk method and integral calculus. Suitable for advanced high school students and calculus enthusiasts.

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