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To solve this problem, we need to use the shell method for finding the volume of the solid. Let's solve each part step by step:

### Part (a)
The formula for the shell method involves the radius, \( x \), and the height, \( f(x) \). From the problem:
- The height of the shell is given by the function \( f(x) = 3x^2 - x^3 \), which is the boundary of the region.

So, the height \( f(x) = 3x^2 - x^3 \).

---

### Part (b)
The volume \( V \) using the shell method is given by:
\[
V = 2\pi \int_0^a x \cdot f(x) \, dx
\]
where \( f(x) = 3x^2 - x^3 \), and \( a \) is the upper limit of integration (where \( y = 3x^2 - x^3 \) intersects \( y = 0 \)).

To find \( a \), solve \( 3x^2 - x^3 = 0 \):
\[
x^2(3 - x) = 0
\]
Thus, \( x = 0 \) or \( x = 3 \). So, \( a = 3 \).

The volume formula becomes:
\[
V = 2\pi \int_0^3 x (3x^2 - x^3) \, dx
\]

---

### Part (c)
Simplify the integrand:
\[
x (3x^2 - x^3) = 3x^3 - x^4
\]
So:
\[
V = 2\pi \int_0^3 (3x^3 - x^4) \, dx
\]

Now compute the integral:
\[
\int_0^3 (3x^3 - x^4) \, dx = \left[ \frac{3x^4}{4} - \frac{x^5}{5} \right]_0^3
\]

Evaluate at the limits:
At \( x = 3 \):
\[
\frac{3(3^4)}{4} - \frac{(3^5)}{5} = \frac{3(81)}{4} - \frac{243}{5} = \frac{243}{4} - \frac{243}{5}
\]

Find a common denominator (20):
\[
\frac{243}{4} = \frac{1215}{20}, \quad \frac{243}{5} = \frac{972}{20}
\]
\[
\frac{243}{4} - \frac{243}{5} = \frac{1215 - 972}{20} = \frac{243}{20}
\]

So:
\[
V = 2\pi \cdot \frac{243}{20} = \frac{486\pi}{20} = \frac{243\pi}{10}
\]

---

### Final Answer:
\[
V = \frac{243\pi}{10}
\]

Let me know if you have any questions or want further details! 

**Questions to explore:**
1. How does the shell method differ from the disk/washer method?
2. Why is \( f(x) = 3x^2 - x^3 \) used for height in this problem?
3. Can the same volume be computed using the disk/washer method? If so, how?
4. What role does symmetry play in simplifying the integration?
5. How would the integral change if the region was rotated about a different axis?

**Tip:** Always double-check the limits of integration when solving problems involving revolutions. Misidentifying limits can lead to significant errors.

Find the volume of the solid obtained by rotating about the y-axis the region bounded by y = 3x^2 - x^3 and y = 0.

Learn how to calculate the volume of a solid obtained by rotating a region about the y-axis using the shell method. This example uses the function y = 3x^2 - x^3, bounded by y = 0, and provides a step-by-step guide including definite integration. Suitable for advanced high school math students.

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