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We are tasked with finding the volume of a solid of revolution, formed by rotating around the \(y\)-axis. The region is bounded by the curves \(y = x\) and \(y = \sqrt{x}\), and the method to be used is the washer (Arandelas) method.

### Step 1: Identify the curves
We have two curves:
- \(y = x\) (which can be rewritten as \(x = y\) when rotating around the \(y\)-axis)
- \(y = \sqrt{x}\) (which can be rewritten as \(x = y^2\))

### Step 2: Washer method
The volume \(V\) of the solid of revolution when rotating around the \(y\)-axis using the washer method is given by the formula:

\[
V = \pi \int_{a}^{b} \left[ \left( \text{outer radius}(y) \right)^2 - \left( \text{inner radius}(y) \right)^2 \right] dy
\]

Here, the outer radius corresponds to the curve \(x = y\) (from \(y = x\)), and the inner radius corresponds to the curve \(x = y^2\) (from \(y = \sqrt{x}\)).

### Step 3: Limits of integration
The curves intersect when \(y = x = \sqrt{x}\), meaning \(y = 1\). Therefore, the bounds for \(y\) are from \(y = 0\) to \(y = 1\).

### Step 4: Set up the integral
The outer radius is \(x = y\), and the inner radius is \(x = y^2\). The volume is then given by:

\[
V = \pi \int_{0}^{1} \left( y^2 - (y^2)^2 \right) dy
\]

This simplifies to:

\[
V = \pi \int_{0}^{1} \left( y^2 - y^4 \right) dy
\]

### Step 5: Evaluate the integral
Now, we compute the integral:

\[
V = \pi \left[ \frac{y^3}{3} - \frac{y^5}{5} \right]_{0}^{1}
\]

At \(y = 1\), this becomes:

\[
V = \pi \left( \frac{1}{3} - \frac{1}{5} \right) = \pi \left( \frac{5 - 3}{15} \right) = \frac{2\pi}{15}
\]

Thus, the volume is \( \frac{2\pi}{15} \).

### Step 6: Compare with the options
The correct formula among the options is:

\[
\boxed{e. \ V = \pi \int_{0}^{1} \left( y^2 - y^4 \right) dy}
\]

Let me know if you'd like more details or further clarification!

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**Related Questions:**
1. How does the washer method differ from the cylindrical shell method?
2. What happens if we rotate around the x-axis instead of the y-axis?
3. How do you determine the limits of integration when finding volumes of revolution?
4. Can the disk method be used in this problem instead of the washer method?
5. What is the geometric interpretation of the outer and inner radii in this context?

**Tip:** Always check whether you need the washer or disk method when finding volumes of solids of revolution!

Dado el sólido de revolución que rota en torno al eje y, el cual está acotado por las ecuaciones y = x, y = √x. La fórmula del volumen de este sólido usando el método de Arandelas es:

Learn how to calculate the volume of a solid of revolution using the washer method. The region is bounded by y = x and y = √x, rotated around the y-axis. Detailed step-by-step integration process provided.

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