(i) The curve \(y = 2 \sin x\) is a sine wave with amplitude 2, period \(2\pi\), and it oscillates between -2 and 2. The curve completes one full cycle from \(0\) to \(2\pi\).

(ii) To find the number of real roots of the equation \(2\pi \sin x = \pi - x\), we rearrange it to \(2\pi \sin x + x = \pi\). This can be rewritten as \(y = 2\pi \sin x\) and \(y = \pi - x\). The line \(y = \pi - x\) can be rewritten as \(y = 1 - \frac{x}{\pi}\) by dividing through by \(\pi\).

The line \(y = 1 - \frac{x}{\pi}\) passes through the points \((0, 1)\) and \((\pi, 0)\). By sketching both the sine curve and the line, we observe that they intersect at three points within the interval \(0 \leq x \leq 2\pi\), indicating there are 3 real roots.