Volume of the solid formed by rotating a circle about the y-axis and finding the distance of its centre

The diagram shows the circle \((x-d)^2+(y-2)^2=4\), where \(d>2\) and \(x,y\in\mathbb{R}\).

The solid \(S\) is formed when the region inside the circle is turned through \(2\pi\) radians about the \(y\)-axis.

1. Show that \(S\) has volume \(4\pi d\displaystyle\int_0^4\sqrt{4-(y-2)^2}\,\mathrm{d}y\).[6]
2. It is now known that \(S\) has a volume of \(50\pi^2\text{ cm}^3\).
   Find \(d\).[2]