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To find the exact value of the volume of the solid obtained by rotating the region bounded by the given curves about the y-axis, we will apply the **method of cylindrical shells**.

The curves provided are:
- \( y = 3\sqrt{x} \)
- \( x = 0 \) (the y-axis)
- \( y = 5 \)

### Step-by-step process:

1. **Express \(x\) in terms of \(y\)** from the equation \( y = 3\sqrt{x} \):
   \[
   y = 3\sqrt{x} \implies \frac{y}{3} = \sqrt{x} \implies x = \left( \frac{y}{3} \right)^2 = \frac{y^2}{9}
   \]

2. **Set up the volume integral using the method of cylindrical shells**:
   The formula for the volume of a solid of revolution using shells is:
   \[
   V = 2\pi \int_{y_{\text{min}}}^{y_{\text{max}}} \text{radius} \cdot \text{height} \, dy
   \]
   In this case:
   - The radius is \(x = \frac{y^2}{9}\), which is the distance from the y-axis.
   - The height is simply the function describing the curve \(y = 3\sqrt{x}\).
   - The bounds are from \(y = 0\) to \(y = 5\).

   Therefore, the volume integral becomes:
   \[
   V = 2\pi \int_0^5 x \, dy = 2\pi \int_0^5 \frac{y^2}{9} \, dy
   \]

3. **Evaluate the integral**:
   \[
   V = \frac{2\pi}{9} \int_0^5 y^2 \, dy
   \]
   The integral of \(y^2\) is:
   \[
   \int y^2 \, dy = \frac{y^3}{3}
   \]
   Applying this:
   \[
   V = \frac{2\pi}{9} \left[ \frac{y^3}{3} \right]_0^5 = \frac{2\pi}{9} \left( \frac{5^3}{3} \right) = \frac{2\pi}{9} \cdot \frac{125}{3}
   \]
   Simplifying:
   \[
   V = \frac{250\pi}{27}
   \]

Thus, the exact value of the volume is:
\[
V = \frac{250\pi}{27}
\]

Would you like further details on any of these steps or additional clarifications?

### Related Questions:
1. What is the method of cylindrical shells and when is it used?
2. How do you convert between \(x\)-coordinates and \(y\)-coordinates for integration?
3. Can the disc/washer method be applied to this problem, and if so, how?
4. How do you handle curves that are not easily solvable for one variable in terms of another?
5. How does the choice of axis of revolution affect the volume calculation?

### Tip:
Always check whether the region being rotated is bounded, as unbounded regions may require additional considerations or limits.

Find the exact value of the volume of the solid obtained by rotating the region bounded by y = 3√x, x = 0, and y = 5, about the y-axis.

Learn how to calculate the exact volume of a solid obtained by rotating the region bounded by y = 3√x, x = 0, and y = 5 about the y-axis using the cylindrical shells method. The detailed step-by-step solution covers key calculus concepts for students in Grades 11-12 or college-level calculus.

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