Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the​ y-axis.
y equals StartRoot x EndRoot​, y equals 0​, x equals 1
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Part 1
Set up the integral that gives the volume of the solid. 
Integral from 0 to nothing left parenthesis nothing right parenthesis dy
  
  
​(Type exact​ answers.)

To solve this problem, we will use the **method of cylindrical shells** to find the volume of the solid generated when the region \( R \) is revolved about the **y-axis**. The given curves are:

- \( y = \sqrt{x} \) (which means \( x = y^2 \) when inverted),
- \( y = 0 \) (the x-axis), and
- \( x = 1 \) (a vertical line).

The steps to set up the integral are as follows:

### 1. Formula for the Volume using Cylindrical Shells:
The volume \( V \) generated by revolving a region around the y-axis can be found using the formula for cylindrical shells:

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

In this case:
- The **radius** is \( x = y^2 \) (because we're revolving around the y-axis and \( x \) is the distance from the y-axis).
- The **height** is given by \( 1 - x \), where \( x = y^2 \).

### 2. Limits of Integration:
- The region is bounded by \( y = 0 \) and \( y = 1 \) (since when \( x = 1 \), \( y = \sqrt{1} = 1 \)).

### 3. Setting up the Integral:
Now, we can set up the integral for the volume. Substituting the values:

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

This integral gives the volume of the solid generated by revolving the region around the y-axis.

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Would you like to see the detailed steps for solving this integral?

Here are 5 related questions:
1. How would the integral change if the region was revolved around the x-axis instead?
2. What if the upper boundary curve was \( y = x^2 \) instead of \( y = \sqrt{x} \)?
3. Can the disk/washer method be used to solve this problem? How would it differ?
4. What is the geometric interpretation of the cylindrical shell method?
5. How would the limits of integration change if the region was bounded by \( x = 2 \)?

**Tip:** Cylindrical shells are especially useful when the axis of rotation is parallel to the side of the solid.

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the​ y-axis. y = √x, y = 0, x = 1. Set up the integral that gives the volume of the solid. Integral from 0 to (1 - y^2) dy.

Learn how to set up the integral for the volume of the solid generated by revolving the region bounded by y=√x, y=0, and x=1 around the y-axis. This problem explores the cylindrical shell method and involves calculating volume using calculus. Suitable for students studying undergraduate calculus or AP Calculus.

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