(i) To find the \(x\)-coordinate of \(M\), we first differentiate \(y = \\sin^2 2x \\cos x\) using the product rule:

\(\frac{dy}{dx} = 4 \\sin 2x \\cos 2x \\cos x - \\sin^2 2x \\sin x\).

Set the derivative to zero and use a double angle formula to simplify:

\(10 \\cos^3 x = 6 \\cos x\).

Solve for \(x\) to obtain \(x = 0.685\).

(ii) Using the substitution \(u = \\sin x\), express the integral in terms of \(u\) and \(du\):

\(du = \\pm \\cos x \, dx\).

The integral becomes \(\int 4u^2 (1-u^2) \, du\).

Use limits \(u = 0\) and \(u = 1\) to evaluate the integral:

\(\frac{8}{15}\) (or 0.533).