https://r2.erweima.ai/mathbot/image/1726607570501f4li2j3r.jpg

The problem involves finding the volume generated by rotating the shaded region bounded by the curves \( f(x) = -x^2 + 18 \) and \( g(x) = x^2 \) about the y-axis. Here's the step-by-step process:

### Step 1: Find the intersection points
The curves intersect where \( f(x) = g(x) \), so we solve the equation:

\[
-x^2 + 18 = x^2
\]

\[
18 = 2x^2 \quad \Rightarrow \quad x^2 = 9 \quad \Rightarrow \quad x = 3
\]

Thus, the region is bounded between \( x = 0 \) and \( x = 3 \).

### Step 2: Set up the integral
Since we are rotating around the y-axis, we will use the method of cylindrical shells. The volume of a shell at a distance \( x \) from the y-axis is given by:

\[
V = 2\pi \int_{0}^{3} x \left( f(x) - g(x) \right) \, dx
\]

Substituting the functions \( f(x) = -x^2 + 18 \) and \( g(x) = x^2 \):

\[
V = 2\pi \int_{0}^{3} x \left( (-x^2 + 18) - x^2 \right) \, dx
\]

Simplifying the integrand:

\[
V = 2\pi \int_{0}^{3} x \left( -2x^2 + 18 \right) \, dx
\]

\[
V = 2\pi \int_{0}^{3} \left( -2x^3 + 18x \right) \, dx
\]

### Step 3: Integrate
Now we integrate term by term:

\[
\int -2x^3 \, dx = -\frac{2x^4}{4} = -\frac{x^4}{2}
\]

\[
\int 18x \, dx = 9x^2
\]

Thus, the integral becomes:

\[
V = 2\pi \left[ \left( -\frac{x^4}{2} + 9x^2 \right) \right]_{0}^{3}
\]

### Step 4: Evaluate the bounds
Substitute \( x = 3 \) into the expression:

\[
\left( -\frac{(3)^4}{2} + 9(3)^2 \right) = \left( -\frac{81}{2} + 81 \right) = -40.5 + 81 = 40.5
\]

At \( x = 0 \), the expression evaluates to 0. So the volume is:

\[
V = 2\pi \times 40.5 = 81\pi
\]

### Final Answer:
The volume generated by rotating the region around the y-axis is \( 81\pi \), which matches the result given in the image.

Do you want further details or have any questions?

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Here are 5 related questions to expand on this topic:

1. How would the volume change if the region were rotated about the x-axis instead of the y-axis?
2. What would be the volume if the functions were \( f(x) = -x^2 + 20 \) and \( g(x) = x^2 \)?
3. Can you compute the area of the shaded region before rotation?
4. What if the rotation were around the line \( x = 1 \)? How would the integral change?
5. How do cylindrical shells compare to using the disk method in this problem?

**Tip:** Always ensure that you correctly identify the bounds of integration when setting up integrals for volume problems.

Find the volume generated by rotating the shaded region bounded by the curves f(x) = -x^2 + 18 and g(x) = x^2 around the y-axis.

Calculate the volume generated by rotating the region bounded by f(x) = -x^2 + 18 and g(x) = x^2 around the y-axis using the cylindrical shell method. The solution involves finding the intersection points and integrating to compute the volume, resulting in 81π.

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