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 xy=x​,

y equals 9 xy=9x​,

y equals 27y=27

y equals 9 xy=9x

y equals xy=x

y equals 27y=27

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2727

Upper RR

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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∫0enter your response hereenter your response here dy

​(Type exact​ answers.)

To solve this problem, we need to compute the volume of the solid generated by revolving the region \( R \) about the \( y \)-axis using the method of cylindrical shells or the method of disks/washers. Based on the equations provided:

### The equations defining the region \( R \) are:
1. \( y = x \)
2. \( y = 9x \)
3. \( y = 27 \)
4. \( x = 0 \) (this is implied since the region is bounded between \( x = 0 \) and the points where the curves meet \( y = 27 \)).

We'll use the **shell method** since the region is being revolved around the \( y \)-axis.

### Step 1: Express \( x \) in terms of \( y \).
- For \( y = x \), this gives \( x = y \).
- For \( y = 9x \), this gives \( x = \frac{y}{9} \).

### Step 2: Set up the integral.
Using the method of cylindrical shells, the volume of a shell is given by:

\[
V = \int_{y_1}^{y_2} 2\pi \cdot \text{radius} \cdot \text{height} \, dy
\]

- The radius of a shell is simply the \( x \)-coordinate, which is \( x = \frac{y}{9} \) and \( x = y \).
- The height of the shell is \( x_{\text{right}} - x_{\text{left}} \), where \( x_{\text{right}} = \frac{y}{9} \) and \( x_{\text{left}} = 0 \).
- The limits of integration will be from \( y = 0 \) to \( y = 27 \).

Thus, the volume \( V \) is given by the integral:

\[
V = 2\pi \int_0^{27} \left( y - \frac{y}{9} \right) \, dy
\]

This is the setup for the integral. If you'd like, I can solve the integral next.

Would you like to proceed with solving the integral or need any further clarification? 

### Follow-up Questions:
1. How do we apply the shell method when revolving around different axes?
2. What is the difference between the shell method and the washer method?
3. How do the limits of integration affect the volume calculation?
4. Can the volume be calculated using the disk/washer method instead for this problem?
5. How do transformations of curves change the region to be revolved?

**Tip**: When setting up volume integrals for revolved solids, always check if cylindrical shells or disks/washers provide a more straightforward approach based on the axis of revolution.

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 = 9x, y = 27, x = 0.

Learn how to set up and compute the volume of a solid formed by revolving the region bounded by the curves y = x, y = 9x, and y = 27 about the y-axis. This example uses the shell method and is suitable for students studying calculus.

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