Let S be the solid obtained by rotating the region shown in the figure below about the y-axis.
Do you think this method is preferable to using washers? Explain your reasoning.


To solve the problem of finding the volume of the solid obtained by rotating the region about the \( y \)-axis, you have two common methods: **cylindrical shells** and **washers (disks)**. Let’s break down when and why one method might be preferable to the other.

1. **Cylindrical Shells Method**:
    - This method is often used when the region is easier to describe as a function of \( x \), but you're rotating around the \( y \)-axis.
    - In this case, for each small slice parallel to the \( y \)-axis, you form a cylindrical shell, and the volume of the shell is calculated.
    - The formula for the volume using shells is:
      \[
      V = 2\pi \int_a^b x \cdot f(x) \, dx
      \]
      where \( f(x) \) is the height of the shell and \( x \) is the distance from the axis of rotation (the \( y \)-axis).
    - This method is particularly advantageous if the region’s boundaries are expressed as functions of \( x \) and it is difficult or messy to solve for \( x \) in terms of \( y \).

2. **Washers Method**:
    - This method works by taking thin slices of the solid perpendicular to the axis of rotation (disks or washers) and calculating the volume of each slice.
    - If rotating around the \( y \)-axis, the integral would be in terms of \( y \).
    - The formula for the volume using washers is:
      \[
      V = \pi \int_c^d \left[ R_{\text{outer}}(y)^2 - R_{\text{inner}}(y)^2 \right] dy
      \]
      where \( R_{\text{outer}}(y) \) and \( R_{\text{inner}}(y) \) represent the radii of the outer and inner curves as functions of \( y \).
    - Washers might be preferable when the region is easier to describe as functions of \( y \) (i.e., when the curves are given as \( x \)-in-terms-of-\( y \) functions).

### Which Method is Preferable?
- **If the curves are described as functions of \( x \) (i.e., \( y = f(x) \)), then the cylindrical shells method is usually preferable**, because it avoids the need to re-express the curves as functions of \( y \). It allows for more straightforward setup and calculation.
- **If the curves are given as functions of \( y \) (i.e., \( x = f(y) \)), then the washers method is often easier**.

However, without the exact shape of the region or the equations for the boundaries, we cannot definitively say which method is better in this specific case. Generally, cylindrical shells are preferred when the region's description is easier in terms of \( x \) and rotating around the \( y \)-axis.

Would you like me to explain more details about one of the methods? 

Here are some questions to explore further:
1. Can you explain how the cylindrical shells method works step by step?
2. How do you determine the limits of integration in the washers method?
3. What are the differences between using disks and washers in volume calculations?
4. Can you provide an example problem using the washers method for a rotation around the \( y \)-axis?
5. How does the radius of a shell or washer affect the total volume?

**Tip**: When choosing between methods, always examine the function forms and axis of rotation to simplify the integral setup.

Let S be the solid obtained by rotating the region shown in the figure below about the y-axis. Do you think this method is preferable to using washers? Explain your reasoning.

Understand the methods to calculate the volume of a solid obtained by rotating a region about the y-axis. This solution explores the cylindrical shells and washers methods, providing step-by-step explanations and key formulas. Ideal for undergraduate calculus students.

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