(i) To find the coordinates of \(P\), set \(y = 3\sqrt{x} = x\). Solving for \(x\), we have:

\(3\sqrt{x} = x\)

\(3 = \sqrt{x}\)

\(x = 9\)

Thus, \(y = x = 9\), so \(P\) is (9, 9).

(ii) To find the area of the shaded region, calculate the area under the curve \(y = 3\sqrt{x}\) from \(x = 0\) to \(x = 9\), and subtract the area under the line \(y = x\) over the same interval.

The area under the curve \(y = 3\sqrt{x}\) is given by:

\(\int_0^9 3\sqrt{x} \, dx = \int_0^9 3x^{1/2} \, dx\)

\(= \left[ 3 \cdot \frac{2}{3} x^{3/2} \right]_0^9\)

\(= \left[ 2x^{3/2} \right]_0^9\)

\(= 2(9^{3/2}) - 2(0^{3/2})\)

\(= 2(27)\)

\(= 54\)

The area under the line \(y = x\) is the area of a right triangle with base and height of 9:

\(\frac{1}{2} \times 9 \times 9 = 40.5\)

Thus, the area of the shaded region is:

\(54 - 40.5 = 13.5\)