Problem

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1

Figure 1 shows a sketch of the curve with equation

$$y = \cos 2x \qquad 0 \le x \le \frac{\pi}{4}$$

The curve is rotated through $2\pi$ radians about the x-axis.

(a) Show that the area of the curved surface generated is given by

$$S = 2\pi \int_0^{\pi/4} \cos 2x \sqrt{1 + 4\sin^2 2x}\,dx$$

(b) Hence, using the substitution $2\sin 2x = \sinh \theta$, show that

$$S = \frac{\pi}{4}\left(\ln(a+\sqrt{b}) + a\sqrt{b}\right)$$

where $a$ and $b$ are integers to be determined.