Find the exact area of the region bounded by the curve \(y = 1 + \mathrm { e } ^ { 2 x - 1 }\), the \(x\)-axis and the lines \(x = \frac { 1 } { 2 }\) and \(x = 2\).
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The diagram shows the curve \(y = \frac { \mathrm { e } ^ { 2 x } } { \sin 2 x }\) for \(0 < x < \frac { 1 } { 2 } \pi\), and its minimum point \(M\). Find the exact \(x\)-coordinate of \(M\).