What is the region in the first quadrant bounded by the graph of \(y=e^{x/2}\) and the line \(x=4\) ?

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The area of the region can be found using:

$$ \int\limits_0^4 e^{\frac{x}{2}}   dx $$ $$ = 2\Big[ e^{\frac{x}{2}} \Big]_0^4 $$ $$ = 2(e^{\frac{4}{2}}-e^{\frac{0}{2}}) $$ $$ = \boxed{2e^2-2} $$