https://r2.erweima.ai/mathbot/image/17241976416892728evwv.jpg

To solve the problem presented in the image, I will first extract the necessary information.

### Problem Statement:
The image depicts a problem asking to find the volume of the solid generated by rotating a certain region around the y-axis. The region is bounded by a curve, the y-axis, and a horizontal line.

### Given Data:
- The curve equation seems to be a quadratic function, possibly \( y = ax^2 + bx + c \).
- There's a visual representation of the solid of revolution with marked boundaries.
- A table provides numerical values for some constants or parameters.

### Step-by-Step Solution:

1. **Identify the curve and the limits of integration:**
   The curve equation appears to be quadratic. Assuming the equation is \( y = ax^2 + bx + c \) (though it might be more specific, such as \( y = x^2 \) depending on the image resolution).

2. **Set up the integral for the volume:**
   The volume \( V \) of the solid generated by rotating the curve around the y-axis can be calculated using the method of disks or washers. The general formula is:
   \[
   V = \pi \int_{y_1}^{y_2} [R(y)]^2 \, dy
   \]
   where \( R(y) \) is the distance from the axis of rotation (the y-axis) to the curve.

   For a curve given by \( y = f(x) \), we need to express \( x \) as a function of \( y \), i.e., \( x = g(y) \), and then square it to get \( [R(y)]^2 \).

3. **Integrate over the appropriate limits:**
   The limits of integration \( y_1 \) and \( y_2 \) are the y-values where the region starts and ends, which you extract from the points where the curve intersects the boundaries.

4. **Substitute the given values and solve the integral.**

Since I do not have full visibility of the exact functions or constants from the image, please verify the curve's equation and the range of integration for accurate computation. 

Would you like me to proceed with a specific formula, or do you need further details on setting up the problem? Also, if you want to extract and use any specific data from the table, I can assist with that as well. Let me know how you would like to continue.

Learn how to calculate the volume of a solid of revolution formed by rotating a quadratic curve around the y-axis. Step-by-step solution provided with integral calculus methods. Suitable for high school students studying integral calculus and geometric solids.

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