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This problem involves finding the volume of the solid obtained by rotating the given region around the **y-axis**. The curves are given as:

\[
y = x - x^2, \quad y = 0
\]

This is a typical application of the **method of cylindrical shells** for volumes of revolution. Here's how we solve it:

### Step 1: Set up the integral using the method of cylindrical shells
In the cylindrical shells method, the formula for the volume is:

\[
V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
\]

where:
- \(x\) is the radius of the cylindrical shell,
- \(f(x)\) is the height of the shell (which is \(y = x - x^2\) here),
- and the limits of integration \(a\) and \(b\) are the \(x\)-values that bound the region.

In this case, the region is bounded by \(x = 0\) and \(x = 1\), so the limits of integration are from 0 to 1.

### Step 2: Write the integral expression
The volume is given by:

\[
V = 2\pi \int_{0}^{1} x \cdot (x - x^2) \, dx
\]

### Step 3: Simplify and integrate
We can simplify the integrand before integrating:

\[
V = 2\pi \int_{0}^{1} (x^2 - x^3) \, dx
\]

Now integrate term by term:

\[
V = 2\pi \left[ \frac{x^3}{3} - \frac{x^4}{4} \right]_{0}^{1}
\]

### Step 4: Compute the definite integral
Substitute the limits of integration \(x = 1\) and \(x = 0\):

\[
V = 2\pi \left( \frac{1^3}{3} - \frac{1^4}{4} \right)
= 2\pi \left( \frac{1}{3} - \frac{1}{4} \right)
= 2\pi \left( \frac{4}{12} - \frac{3}{12} \right)
= 2\pi \cdot \frac{1}{12}
= \frac{\pi}{6}
\]

### Final Answer:
The volume of the solid obtained by rotating the region about the y-axis is:

\[
V = \frac{\pi}{6}
\]

---

Would you like any further details or explanations? Here are some related questions to explore:

1. What is the method of cylindrical shells and how does it differ from the disk/washer method?
2. How would the integral change if the region were rotated around the x-axis instead?
3. Could you set up the problem using the washer method for rotation about the y-axis?
4. How do you find the limits of integration in problems involving volumes of revolution?
5. What if the curve was shifted up or down? How would that affect the volume?

**Tip:** When using the cylindrical shells method, remember that the radius of each shell corresponds to the distance from the axis of rotation.

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

This detailed solution explores finding the volume of a solid formed by rotating the region bounded by y = x - x^2 and y = 0 around the y-axis using the cylindrical shells method. Learn step-by-step how to set up the integral and compute the definite integral to find the volume.

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