Problems

In the figure, the large circle with centre \(O\) has radius \(4\) and the small circle with centre \(P\) has radius \(1\). The small circle rolls around the inside of the larger one. When \(P\) was on the line \(OA\) (before the small circle began to roll), the point \(B\) was in contact with the point \(A\) on the large circle.

Sketch the curve \(C\) traced by \(B\) as the circle rolls. Show that if we take \(O\) to be the origin of cartesian coordinates and the line \(OA\) to be the \(x\)-axis (so that \(A\) is the point \((4,0)\)) then \(B\) is the point \[ (3\cos\phi+\cos3\phi,3\sin\phi-\sin3\phi). \] It is given that the area of the region enclosed by the curve \(C\) is \[ \int_{0}^{2\pi}x\frac{\mathrm{d}y}{\mathrm{d}\phi}\,\mathrm{d}\phi, \] where \(B\) is the point \((x,y).\) Calculate this area.