Applications of Integration

The region \(R\) is the finite region bounded by the curve \({{y}^{2}}+{{(x-2)}^{4}}=1\).

1. Sketch the curve and indicate clearly the region \(R\).
2. Find the area of \(R\).
3. The curve \(y={{(x-1)}^{2}}\) divides \(R\) into two regions. Find the exact volume of the solid of revolution formed when the smaller region is rotated through \(4\) right angles about the \(x\)-axis.