Use the washer method to determine the volume of the solid formed when the region bounded by y=x and y=x3 is rotated about the y-axis over the interval 0≤x≤1.

To determine the volume of the solid formed by rotating the region bounded by the curves \( y = x \) and \( y = x^3 \) around the y-axis, we will use the **washer method**.

### Steps:
1. **Express the curves in terms of \( x \):**
   - The first curve is \( y = x \), so \( x = y \) when expressed in terms of \( y \).
   - The second curve is \( y = x^3 \), so \( x = y^{1/3} \) when expressed in terms of \( y \).

2. **Determine the limits of integration:**
   The bounds for \( y \) are from 0 to 1, as the given curves intersect at \( x = 0 \) and \( x = 1 \), corresponding to \( y = 0 \) and \( y = 1 \).

3. **Set up the washer method formula:**
   The volume \( V \) of a solid of revolution using the washer method is given by:

   \[
   V = \pi \int_{y_1}^{y_2} \left( R_{\text{outer}}(y)^2 - R_{\text{inner}}(y)^2 \right) dy
   \]
   Where:
   - \( R_{\text{outer}}(y) \) is the outer radius, which corresponds to the curve \( y = x \) (i.e., \( x = y \)).
   - \( R_{\text{inner}}(y) \) is the inner radius, which corresponds to the curve \( y = x^3 \) (i.e., \( x = y^{1/3} \)).

4. **Apply the formula:**
   The outer radius is \( R_{\text{outer}}(y) = y^{1/3} \), and the inner radius is \( R_{\text{inner}}(y) = y \). The volume integral becomes:

   \[
   V = \pi \int_0^1 \left( y^{2/3} - y^2 \right) dy
   \]

5. **Solve the integral:**

   \[
   V = \pi \int_0^1 \left( y^{2/3} - y^2 \right) dy
   \]

   First, find the integral of each term:

   \[
   \int y^{2/3} dy = \frac{3}{5} y^{5/3}
   \]
   and
   \[
   \int y^2 dy = \frac{y^3}{3}
   \]

   Therefore, the volume becomes:

   \[
   V = \pi \left[ \frac{3}{5} y^{5/3} - \frac{y^3}{3} \right]_0^1
   \]

6. **Evaluate at the bounds:**

   At \( y = 1 \):
   \[
   \left( \frac{3}{5}(1)^{5/3} - \frac{1^3}{3} \right) = \frac{3}{5} - \frac{1}{3}
   \]
   Find the common denominator:
   \[
   \frac{3}{5} - \frac{1}{3} = \frac{9}{15} - \frac{5}{15} = \frac{4}{15}
   \]

   At \( y = 0 \), both terms are zero.

   So the total volume is:
   \[
   V = \pi \times \frac{4}{15} = \frac{4\pi}{15}
   \]

### Final Answer:
The volume of the solid formed by rotating the region bounded by \( y = x \) and \( y = x^3 \) around the y-axis is:
\[
V = \frac{4\pi}{15}
\]

Would you like further details or explanations on any of these steps?

### Related Questions:
1. How does the washer method differ from the shell method?
2. What would change if the region was rotated around the x-axis instead of the y-axis?
3. How do we handle cases where the boundaries are not simple functions of x or y?
4. How would you find the volume if the curves were given parametrically?
5. How do we interpret the physical meaning of a volume of revolution in real-world contexts?

### Tip:
Always ensure that you correctly identify the outer and inner radii when setting up the washer method to avoid errors in the integral setup.

Use the washer method to determine the volume of the solid formed when the region bounded by y=x and y=x^3 is rotated about the y-axis over the interval 0≤x≤1.

Learn how to determine the volume of a solid formed by rotating the region bounded by y=x and y=x^3 around the y-axis over the interval 0≤x≤1. This solution covers key calculus concepts such as the washer method and integration, providing a step-by-step approach. Ideal for students at an undergraduate calculus level.

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