To find the volume of the solid generated by revolving the region $R$ bounded by the curves $y = x - x^2$ and $y = 0$ around the y-axis using the shell method, we follow these steps:

1. Identify the region and the curves:

   • The region $R$ is bounded by $y = x - x^2$ and $y = 0$.
   • We also need the interval over which $x$ varies. To find this interval, we solve $y = 0$ for $x$: $x - x^2 = 0 \implies x(x-1) = 0 \implies x = 0 \text{ or } x = 1.$
   • Therefore, $x$ varies from 0 to 1.

2. Set up the shell method formula:

   • For the shell method, the volume $V$ is given by: $V = 2\pi \int_{a}^{b} \text{(radius)} \times \text{(height)} \, dx$
   • Here, the radius of each cylindrical shell is the distance from the y-axis, which is $x$.
   • The height of each shell is the function value $y$, which is $x - x^2$.

3. Integrate the volume:

   • The volume integral becomes: $V = 2\pi \int_{0}^{1} x (x - x^2) \, dx$
   • Simplify the integrand: $V = 2\pi \int_{0}^{1} (x^2 - x^3) \, dx$
   • Integrate term by term: $\int_{0}^{1} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{1} = \frac{1}{3} - 0 = \frac{1}{3}$ $\int_{0}^{1} x^3 \, dx = \left[ \frac{x^4}{4} \right]_{0}^{1} = \frac{1}{4} - 0 = \frac{1}{4}$
   • Combine the results: $V = 2\pi \left( \frac{1}{3} - \frac{1}{4} \right) = 2\pi \left( \frac{4}{12} - \frac{3}{12} \right) = 2\pi \left( \frac{1}{12} \right) = \frac{\pi}{6}$

Therefore, the volume of the solid generated by revolving the region $R$ around the y-axis is: $V = \frac{\pi}{6}$

Would you like further details or have any questions?

Related Questions:

1. How do you find the volume of a solid using the washer method?
2. What are the differences between the shell method and the washer method?
3. Can you use the shell method to find the volume of solids of revolution around the x-axis?
4. How do you determine the height of the shell in the shell method?
5. What are common mistakes to avoid when using the shell method?
6. How does the shell method change if the region is revolved around a vertical line other than the y-axis?
7. How do you set up the integral for the washer method?
8. Can the shell method be used for regions bounded by more complicated functions?

Tip:

When using the shell method, always ensure that the radius and height are correctly identified relative to the axis of rotation. This will help avoid common mistakes in setting up the integral.