draw an outline of the solid and find the volume using the slicing method. Consider that the region is enclosed by the curves y is equal to x^(2) and y is equal to 9.

Let's break this down step by step to solve the problem.

### 1. Understand the Geometry

You are given two curves:

- \( y = x^2 \)
- \( y = 9 \)

These curves enclose a region in the \( xy \)-plane. The curve \( y = x^2 \) is a parabola, and the line \( y = 9 \) is a horizontal line. The region we are interested in lies between these curves.

To begin, we need to determine the points of intersection between the two curves. We can do this by setting:

\[
x^2 = 9
\]

Taking the square root of both sides:

\[
x = \pm 3
\]

Thus, the points of intersection are \( x = -3 \) and \( x = 3 \).

### 2. Sketch the Region

The region enclosed by the curves is the area between the parabola \( y = x^2 \) and the line \( y = 9 \), from \( x = -3 \) to \( x = 3 \).

Now let's visualize the solid. When we revolve the enclosed region around the x-axis (which is typical for this kind of slicing method problem), we will create a solid of revolution.

### 3. Find the Volume Using the Slicing Method

Using the slicing method (also known as the disk method), we slice the solid perpendicular to the x-axis.

The volume of a thin slice (a disk) at a point \( x \) is given by:

\[
\text{Volume of a slice} = \pi \cdot \left(\text{radius}\right)^2 \cdot \text{thickness}
\]

In this case, the radius of the disk at point \( x \) is the distance from the x-axis to the curve \( y = x^2 \), which is simply \( y \). Thus, the radius of each disk is:

\[
\text{Radius} = 9 - x^2
\]

The thickness of the disk is a small change in \( x \), denoted by \( dx \). Hence, the volume of the entire solid can be computed by integrating from \( x = -3 \) to \( x = 3 \):

\[
V = \int_{-3}^{3} \pi \cdot \left(9 - x^2\right)^2 \, dx
\]

I will compute this volume now.The volume of the solid formed by revolving the region enclosed by the curves \( y = x^2 \) and \( y = 9 \) around the x-axis is:

\[
V = \frac{1296\pi}{5} \, \text{cubic units}
\]

Would you like further clarification on any of these steps or additional details on the slicing method? Here are some related questions that might expand your understanding:

1. What would happen if the region was rotated around the y-axis instead of the x-axis?
2. How would the method change if the enclosed region was rotated around a different line, such as \( y = 9 \)?
3. How do we handle volumes for regions enclosed by different types of curves (e.g., trigonometric curves)?
4. Can the washer method be used for this problem, and how would the steps differ?
5. How can this method be extended to find the surface area of the solid?

**Tip:** Visualizing the 2D region before rotation helps in understanding the symmetry and properties of the resulting solid.

Draw an outline of the solid and find the volume using the slicing method. Consider that the region is enclosed by the curves y = x^2 and y = 9.

This calculus problem demonstrates how to find the volume of a solid by revolving the region enclosed by y = x^2 and y = 9 around the x-axis using the slicing method. Suitable for Grades 11-12 and college calculus students, learn how to apply the disk method with step-by-step integration.

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