Area under polynomial curve

\includegraphics{figure_5} Figure 5 shows a sketch of the curve \(C\) with equation \(y = (x - 2)^2(x + 3)\) The region \(R\), shown shaded in Figure 5, is bounded by \(C\), the vertical line passing through the maximum turning point of \(C\) and the \(x\)-axis. Find the exact area of \(R\). *(Solutions based entirely on graphical or numerical methods are not acceptable.)* [6]

The diagram below represents the graph of the function \(y = (2x - 5)^4 - 1\) \includegraphics{figure_7}

1. Find the intersections of this graph with the \(x\) axis. [1]
2. Hence find the exact value of the area bounded by the curve and the \(x\) axis. [4]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is \(48\) units\(^2\), determine the \(y\)-coordinate of \(P\). [7]