(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:
\(54 - 40.5 = 13.5\)
