Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = sin−1(x),   y = pi/2
,   x = 0;
about y = 2. in terms of dy and using cylindrical shells

To set up the integral for the volume of the solid obtained by rotating the region bounded by the curves \( y = \sin^{-1}(x) \), \( y = \frac{\pi}{2} \), and \( x = 0 \) about the line \( y = 2 \) using the method of cylindrical shells in terms of \( dy \), we proceed as follows:

### 1. **Identify the boundaries and functions:**

- The curve \( y = \sin^{-1}(x) \) can be rewritten as \( x = \sin(y) \).
- The other boundaries are \( y = \frac{\pi}{2} \) and \( x = 0 \).

### 2. **Understand the cylindrical shell method:**

The formula for the volume \( V \) using cylindrical shells is:

\[
V = 2\pi \int_{a}^{b} (\text{radius}) \times (\text{height}) \, dy
\]

Where:
- **Radius** is the distance from the axis of rotation (\( y = 2 \)) to the element \( y \).
- **Height** is the value of the function describing the boundary, which in this case is \( x = \sin(y) \).

### 3. **Determine the radius and height:**

- The **radius** of the cylindrical shell at a point \( y \) is the vertical distance from \( y = 2 \) to \( y \). Thus, the radius is \( |2 - y| \) or simply \( 2 - y \) (since \( y \) is less than or equal to 2 within the given limits).
- The **height** of the cylindrical shell is the horizontal distance from \( x = 0 \) to the curve \( x = \sin(y) \).

### 4. **Determine the bounds of integration:**

The bounds for \( y \) are from \( y = 0 \) to \( y = \frac{\pi}{2} \).

### 5. **Set up the integral:**

The volume \( V \) using cylindrical shells is given by:

\[
V = 2\pi \int_{0}^{\frac{\pi}{2}} (2 - y) \cdot \sin(y) \, dy
\]

This is the integral that represents the volume of the solid using the cylindrical shell method.

Would you like more details on how to evaluate this integral or have any questions?

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### Related Questions:
1. What is the difference between the washer method and the cylindrical shell method?
2. How do you evaluate the integral for the volume once it's set up using cylindrical shells?
3. Can you explain why the radius in the cylindrical shell method is \( 2 - y \)?
4. What if the region were rotated about a different line, say \( y = 3 \)?
5. How would the integral change if the region were rotated about the x-axis instead?

### Tip:
When using the cylindrical shell method, remember that the radius is always the distance from the axis of rotation to the element you're integrating, and the height corresponds to the function you're rotating. This distinction is crucial for correctly setting up the integral.

Learn how to set up, but not evaluate, an integral for the volume of a solid obtained by rotating the region bounded by y = sin^{-1}(x), y = \frac{\pi}{2}, x = 0 about y = 2 using cylindrical shells and in terms of dy.

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