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To solve this problem using the shell method, let's break down the given information.

The question provides:
1. Curves: \( y = x^2 \) and \( y = 3 - 2x \)
2. Limits: \( x = 0 \) and \( x \geq 0 \)
3. Rotation axis: the \( y \)-axis.

### Step 1: Find the Points of Intersection
To determine the bounds for \( x \), find the points where the two curves intersect by setting \( y = x^2 = 3 - 2x \).

\[
x^2 + 2x - 3 = 0
\]

Solving this quadratic equation:
\[
(x + 3)(x - 1) = 0
\]

Thus, \( x = -3 \) and \( x = 1 \). Since we are only considering \( x \geq 0 \), the relevant intersection point is \( x = 1 \).

### Step 2: Set Up the Shell Method Integral
The shell method formula for rotation about the \( y \)-axis is:
\[
V = 2\pi \int_{a}^{b} x \cdot (f(x) - g(x)) \, dx
\]

Here:
- \( f(x) = 3 - 2x \)
- \( g(x) = x^2 \)
- Bounds of integration: \( x = 0 \) to \( x = 1 \)

### Step 3: Substitute and Integrate
\[
V = 2\pi \int_{0}^{1} x \cdot ((3 - 2x) - x^2) \, dx
\]
Simplify the expression inside the integral:
\[
V = 2\pi \int_{0}^{1} x \cdot (3 - 2x - x^2) \, dx
\]
\[
V = 2\pi \int_{0}^{1} (3x - 2x^2 - x^3) \, dx
\]

Now integrate term-by-term:
\[
V = 2\pi \left[ \frac{3x^2}{2} - \frac{2x^3}{3} - \frac{x^4}{4} \right]_{0}^{1}
\]
Evaluating at the limits:
\[
V = 2\pi \left( \frac{3(1)^2}{2} - \frac{2(1)^3}{3} - \frac{(1)^4}{4} \right)
\]
\[
V = 2\pi \left( \frac{3}{2} - \frac{2}{3} - \frac{1}{4} \right)
\]

To combine these terms, find a common denominator:
\[
V = 2\pi \left( \frac{18}{12} - \frac{8}{12} - \frac{3}{12} \right)
\]
\[
V = 2\pi \cdot \frac{7}{12}
\]
\[
V = \frac{7\pi}{6}
\]

### Final Answer
The volume is \( \frac{7\pi}{6} \), which matches option **(c)**.

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Would you like more details on the steps, or do you have further questions?

### Related Questions:
1. How does the shell method differ from the disk method for volume calculation?
2. What would change if the region were rotated around the \( x \)-axis instead?
3. Can the volume be solved using the washer method for this problem?
4. What are the applications of the shell method in real-world scenarios?
5. How would the setup change if we had different limits of \( x \)?

### Tip:
For volume problems, carefully sketch the region and rotation axis—it helps to set up the integral correctly.

Use the shell method to find the volume of the solid generated by revolving the regions bounded by the curves and lines about the y-axis. y = x^2, y = 3 - 2x, x = 0, for x ≥ 0.

Learn how to find the volume of a solid generated by revolving the region bounded by y = x^2, y = 3 - 2x, and x = 0 around the y-axis using the shell method. This problem covers calculus concepts of integration and the shell method, ideal for college-level calculus students.

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