We are given a region $R$ in the first quadrant, symmetric with respect to the line $y = 3$, and its area is $\frac{5}{9}$ square units. The region is rotated around the line $y = -5$, and we need to find the volume of the resulting solid.

Step 1: Analyze the symmetry

The region $R$ is symmetric with respect to the line $y = 3$. This means that if we have a function $f(x)$ defining the upper half of the region above $y = 3$, the lower half would be reflected across $y = 3$, i.e., $6 - f(x)$ for the part below.

Step 2: Set up the volume formula

Since the solid is generated by rotating around a horizontal line $y = -5$, we can use the method of cylindrical shells or the washer method. The washer method is easier to apply here, as we rotate the region around a line parallel to the axis of the region's symmetry.

The formula for the volume generated by rotating a region around a horizontal line is:

$V = \pi \int_{a}^{b} \left[ (R_{\text{outer}}(x))^2 - (R_{\text{inner}}(x))^2 \right] dx$

Where:

• $R_{\text{outer}}(x)$ is the distance from the outer curve to the axis of rotation.
• $R_{\text{inner}}(x)$ is the distance from the inner curve to the axis of rotation.

Step 3: Distances to the axis of rotation

The axis of rotation is $y = -5$, and the region is symmetric about $y = 3$. Therefore:

• The outer radius (distance from the top of the region) is $R_{\text{outer}}(x) = f(x) + 5$.
• The inner radius (distance from the bottom of the region, assuming it's symmetric) is $R_{\text{inner}}(x) = 6 - f(x) + 5 = 11 - f(x)$.

Step 4: Volume of revolution

Now, we substitute the expressions for the radii into the volume integral. Let the region span from $x = a$ to $x = b$. The volume is:

$V = \pi \int_a^b \left[ (f(x) + 5)^2 - (11 - f(x))^2 \right] dx$

At this point, we need either an explicit function for $f(x)$ or further information about the shape of the region. However, knowing that the area of the region is $\frac{5}{9}$ square units and the problem is symmetric suggests that the specific geometry of the region is key to solving this. But as a next step, we would evaluate this integral, given $f(x)$, to compute the exact volume.

Do you have any specific details about the function $f(x)$, or would you like to proceed with approximations based on symmetry?

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Follow-up questions:

1. How would the volume change if the region was rotated around a different line, such as $y = 3$?
2. What would happen to the volume if the region's area were doubled while keeping the same symmetry?
3. How does the washer method compare to the cylindrical shells method for this problem?
4. What is the significance of the line of symmetry in determining the volume of revolution?
5. Could this region's symmetry with respect to $y = 3$ affect how we split the integral?

Tip: When rotating regions around lines other than the axis, carefully calculate the radius as the distance from the curve to the axis of rotation, not just the height of the function.