(a)[4]
Find the exact area of the region enclosed by the curve $y = 1 + e^{2x-1}$, the $x$-axis and the lines $x = \frac{1}{2}$ and $x = 2$.
(b)[5]
The diagram displays the curve $y = \frac{e^{2x}}{\sin 2x}$ for $0 < x < \frac{1}{2}\pi$, together with its minimum point $M$. Find the exact $x$-coordinate of $M$.