The diagram shows part of the curve \(y = \sqrt{4x+1} + \frac{9}{\sqrt{4x+1}}\) and the minimum point \(M\).

(i) Find expressions for \(\frac{dy}{dx}\) and \(\int y \, dx\).

(ii) Find the coordinates of \(M\).

The shaded region is bounded by the curve, the \(y\)-axis and the line through \(M\) parallel to the \(x\)-axis.

(iii) Find, showing all necessary working, the area of the shaded region.