2025 TJC Promo Q11

A school builds a flat marble slab as a monument to commemorate its \(60^{\text{th}}\) school anniversary. The marble slab is made in the shape of the shaded region enclosed by the curve \(C\) with equation \(\frac{x^2}{9}+4(y-1)^2=1\), the vertical lines \(x=-3\) and \(x=3\) and the \(x\)-axis, as shown in Figure 1.

1. Using the substitution \(x=3\sin\theta\), find the exact area of \(R\).[6]

The school builds a water fountain as a second monument. The base of the water fountain is the solid formed when \(R\) is rotated through \(\pi\) radians about the \(y\)-axis, as shown in Figure 2.

3. Without using a graphing calculator, find the exact volume of the base of the water fountain.[4]