2024 YIJC P2 Q3

The region \(R\) is bounded by the curve \(y=\frac{1}{1+\sqrt{x}}\), the \(x\)-axis and the lines \(x=1\) and \(x=4\). \(R\) is rotated about the \(x\)-axis through \(2\pi \) radians. Using the substitution \(u=\sqrt{x}\), find the exact volume of the solid generated.[6]

The region \(L\) is bounded by the curve \(y=\frac{1}{1+\sqrt{x+a}}\), the \(x\)-axis and the lines \(x=3\) and \(x=6\). Given that the volume of the solid generated when \(L\) is rotated about the \(x\)-axis through \(2\pi \) radians is the same as the volume calculated in part (a), state the value of \(a\).[1]