Answer
$e^{\frac{x}{2}} -1$
Work Step by Step
from $L$ = $\int_{{\,a}}^{{\,b}}$$\sqrt{1+(\frac{dy}{dx})^{2}}$ $dx$
and $L$ = $\int_{{\,0}}^{{\,1}}$$\sqrt{1+\frac{1}{4}e^{x}}$ $dx$
so
$(\frac{dy}{dx})^{2}$ = $\frac{1}{4}e^{x}$
$\frac{dy}{dx}$ = $\frac{1}{2}e^{\frac{x}{2}}$
$y$ = $\int$$\frac{1}{2}e^{\frac{x}{2}}$$dx$
$y$ = $e^{\frac{x}{2}} + C$
Curve through the original so $y(0)$ = $0$
$0$ = $e^{0} + C$
$0$ = $1 + C$
$C$ = $-1$
$y$ = $e^{\frac{x}{2}} -1$
