IB AA HL May 2026 P2 TZA Q9

Consider the curve \((x-d)^2+(y-2)^2=4\) where \(x,y\in\mathbb{R}\) and \(d>2\), shown in the following diagram.

The region enclosed by the curve is rotated through \(2\pi\) radians around the \(y\)-axis to form a solid, \(S\).

1. Show that the volume of \(S\) is given by \(4\pi d∫_0^4\sqrt{4-(y-2)^2}\,\mathrm{d}y\).[6]
2. It is given that the volume of \(S\) is \(50\pi^2\text{ cm}^3\).
   Find the value of \(d\).[2]