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The problem provided in the image is as follows:

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**Your solution should be concise and easy to follow.**

Consider a solid formed by rotating the region bounded by \( y = x^2 \), \( y = 9 \), and \( x = 0 \) about the y-axis.

- **Find the volume of the solid.**
- **Set up an integral to integrate out the volume of the solid.**

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### Step 1: Analyzing the Problem
We are given the following curves and axis of rotation:

1. \( y = x^2 \): This is a parabola opening upwards.
2. \( y = 9 \): This is a horizontal line at \( y = 9 \).
3. \( x = 0 \): This is the y-axis.

The region bounded by these curves will be rotated about the y-axis to form a solid. To find the volume of the solid, we'll use the method of disks or washers.

### Step 2: Expressing \( x \) as a function of \( y \)
Given \( y = x^2 \), we can express \( x \) as a function of \( y \):
\[
x = \sqrt{y}
\]
This will be the radius of the disks when rotating around the y-axis.

### Step 3: Setting up the Integral
The volume of the solid formed by rotating the region about the y-axis can be found using the formula:
\[
V = \pi \int_{a}^{b} [R(y)]^2 \, dy
\]
where \( R(y) \) is the radius of the disk at height \( y \).

In this case:
- The bounds of integration \( a \) and \( b \) are from \( y = 0 \) to \( y = 9 \).
- The radius \( R(y) \) is given by \( R(y) = \sqrt{y} \).

Thus, the integral becomes:
\[
V = \pi \int_{0}^{9} (\sqrt{y})^2 \, dy
\]
Simplifying the integrand:
\[
V = \pi \int_{0}^{9} y \, dy
\]

### Step 4: Calculating the Volume
Now, we calculate the integral:
\[
V = \pi \left[ \frac{y^2}{2} \right]_{0}^{9} = \pi \left( \frac{9^2}{2} - \frac{0^2}{2} \right)
\]
\[
V = \pi \left( \frac{81}{2} \right) = \frac{81\pi}{2}
\]

### Final Answer
The volume of the solid is:
\[
V = \frac{81\pi}{2} \text{ cubic units.}
\]

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Would you like further details or have any questions? Here are some related questions to expand your understanding:

1. How would the volume change if the region were rotated about the x-axis instead of the y-axis?
2. What if the curve was \( y = x^3 \) instead of \( y = x^2 \)? How would that affect the integral?
3. How can the shell method be used to find the volume of the solid in this problem?
4. What is the significance of choosing different axes of rotation in volume calculations?
5. How does the integral setup change if the region is bounded by another curve, such as \( y = 2x + 1 \)?

**Tip:** When dealing with solids of revolution, carefully choose between the disk/washer method and the shell method based on the symmetry of the problem for simpler integration.

Consider a solid formed by rotating the region bounded by y = x^2, y = 9, and x = 0 about the y-axis. Find the volume of the solid. Set up an integral to integrate out the volume of the solid.

Learn how to find the volume of a solid formed by rotating the region bounded by y = x^2, y = 9, and x = 0 about the y-axis. This problem involves the method of disks/washers and integration, suitable for Grades 11-12.

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