The problem describes a three-dimensional volume where the base lies between two parabolas, and the cross-sections perpendicular to the $y$-axis are semi-circles. We are tasked with finding the volume of this solid.

Step-by-Step Solution:

1. Identify the boundaries: The base is the region bounded by the curves:

   • $x = -y^2 + 16y - 43$
   • $x = y^2 - 18y + 101$

   To determine the limits of integration, we need to find the points where these two curves intersect. This happens when $-y^2 + 16y - 43 = y^2 - 18y + 101$.

   Rearranging this equation: $-y^2 + 16y - 43 = y^2 - 18y + 101$ Combine like terms: $-2y^2 + 34y - 144 = 0$ Divide through by $-2$: $y^2 - 17y + 72 = 0$ Solve this quadratic equation using the quadratic formula: $y = \frac{-(-17) \pm \sqrt{(-17)^2 - 4(1)(72)}}{2(1)}$ $y = \frac{17 \pm \sqrt{289 - 288}}{2}$ $y = \frac{17 \pm \sqrt{1}}{2}$ $y = \frac{17 \pm 1}{2}$ So, $y = 9$ and $y = 8$. These are the limits of integration.

2. Determine the diameter of the semi-circular cross-sections: The diameter of each semi-circle is the horizontal distance between the two curves at a given $y$-coordinate. This distance is given by: $D(y) = \left( y^2 - 18y + 101 \right) - \left( -y^2 + 16y - 43 \right)$ Simplifying: $D(y) = y^2 - 18y + 101 + y^2 - 16y + 43$ $D(y) = 2y^2 - 34y + 144$

3. Find the radius of the semi-circular cross-sections: The radius $r(y)$ is half of the diameter: $r(y) = \frac{D(y)}{2} = \frac{2y^2 - 34y + 144}{2} = y^2 - 17y + 72$

4. Determine the area of each semi-circular cross-section: The area of a semi-circle is: $A(y) = \frac{1}{2} \pi r(y)^2$ Substituting $r(y) = y^2 - 17y + 72$: $A(y) = \frac{1}{2} \pi (y^2 - 17y + 72)^2$

5. Set up the integral to find the volume: The volume is the integral of the areas of the semi-circular cross-sections from $y = 8$ to $y = 9$: $V = \int_{8}^{9} A(y) \, dy = \int_{8}^{9} \frac{1}{2} \pi (y^2 - 17y + 72)^2 \, dy$

6. Simplify and calculate: We can now proceed to compute this integral.

Let me compute the exact value for the volume.The volume of the solid is approximately $0.01667 \pi$ cubic units.

Would you like more details on any of the steps? Here are 5 related questions to expand your understanding:

1. How do you calculate the area of a semi-circle?
2. Why do we need to solve for the intersection points of the curves?
3. What is the significance of the radius function in determining the volume?
4. How would the volume change if the cross-sections were full circles instead of semi-circles?
5. Can this method be applied to other types of cross-sectional shapes?

Tip: When solving for the volume of solids with known cross-sectional shapes, always identify the base and the cross-sectional area as functions of a variable.