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We are asked to find the volume of the solid generated by revolving the region around the y-axis. The region is bounded by:

- The parabola \( y = 3x^2 \)
- The x-axis (i.e., \( y = 0 \))
- The line \( x = 1 \)

Since this is a revolution around the y-axis, we can use the *method of cylindrical shells* to calculate the volume.

### Step-by-step solution:

1. **Formula for the volume using cylindrical shells**:
   The formula for the volume of a solid formed by revolving around the y-axis is:
   \[
   V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
   \]
   where \( f(x) \) represents the curve, and \( a \) and \( b \) are the bounds for \( x \).

2. **Bounds and function**:
   - The curve is \( y = 3x^2 \), so \( f(x) = 3x^2 \).
   - The limits of integration are \( x = 0 \) to \( x = 1 \), since the region is bounded on the right by \( x = 1 \) and we are considering the first quadrant.

3. **Set up the integral**:
   Substituting \( f(x) = 3x^2 \) into the formula:
   \[
   V = 2\pi \int_0^1 x \cdot 3x^2 \, dx
   \]
   Simplifying the integrand:
   \[
   V = 2\pi \int_0^1 3x^3 \, dx
   \]

4. **Evaluate the integral**:
   \[
   V = 2\pi \left[ \frac{3x^4}{4} \right]_0^1
   \]
   At \( x = 1 \):
   \[
   \frac{3(1)^4}{4} = \frac{3}{4}
   \]
   At \( x = 0 \):
   \[
   \frac{3(0)^4}{4} = 0
   \]
   So the volume is:
   \[
   V = 2\pi \cdot \frac{3}{4} = \frac{3\pi}{2}
   \]

Thus, the exact volume is:
\[
V = \frac{3\pi}{2}
\]

---

Would you like further details or clarifications? Here are five related questions that might expand your understanding:

1. How would the volume change if the region was revolved around the x-axis instead of the y-axis?
2. Can this method of cylindrical shells be applied to any shape, or are there limitations?
3. What is the difference between the disk/washer method and the method of cylindrical shells?
4. How do we handle cases where the region is bounded by more complex curves?
5. If the limits of integration were different, say \( x = 0 \) to \( x = 2 \), how would the solution change?

**Tip**: For rotational volume problems, always verify if you should use the disk/washer method or the cylindrical shells method, based on the axis of rotation and the region described.

Find the volume of the solid generated by revolving the region bounded by y = 3x^2, the x-axis, and x = 1 around the y-axis.

Learn how to find the volume of a solid formed by revolving the region bounded by the parabola y = 3x^2, the x-axis, and x = 1 around the y-axis using the method of cylindrical shells. Includes step-by-step calculations and key concepts of calculus. Suitable for college-level calculus students.

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