(i) To find the area of the shaded region, integrate with respect to \(y\):

\(A = \int_1^2 \left( y^2 - 1 \right) \frac{1}{3} \, dy\)

\(= \left[ \frac{y^3}{9} - \frac{y}{3} \right]_1^2 = \frac{4}{9}\)

(ii) To find the volume when the region is rotated about the \(x\)-axis:

\(V = \pi \int_0^1 y^2 \, dx = \pi \int_0^1 (3x + 1) \, dx\)

\(= \pi \left[ \frac{3x^2}{2} + x \right]_0^1 = \pi \left( \frac{3}{2} + 1 \right) = \frac{5\pi}{2}\)

Subtract the volume of the cylinder: \(\pi \times 2^2 \times 1 = 4\pi\)

\(\text{Volume} = 4\pi - \frac{5\pi}{2} = 1.5\pi\)

(iii) To find the angle between the tangents, calculate the derivatives:

\(\frac{dy}{dx} = \frac{1}{2}(3x + 1)^{-\frac{1}{2}} \times 3\)

At \(x = 0\), \(m = \frac{3}{2}\). At \(x = 1\), \(m = \frac{3}{4}\).

Calculate the angles: \(\arctan(\frac{3}{2}) \approx 56.3^\circ\) and \(\arctan(\frac{3}{4}) \approx 36.9^\circ\).

The angle between the tangents is \(56.3^\circ - 36.9^\circ = 19.4^\circ\).