. The area bounded by $y=e^x$ , Y-axis and the line y = e is

The given curve is $y=e^x\Rightarrow x={\log }_{e}y$

The area of required region is as below

The area of the shaded region is the area of the region bouned by the curve $x=\log y$ and the lines $y=1,y=e$ and $y-$ axis

Hence, the required area is 

$A={\int }_1^e\log xdx$

Use the integraion by parts to evaluate the above integral

$\begin{array}{rcl}A& =& {\int }_1^{e}1·\log xdx\\ & =& {\left[x·\log x\right]}_1^e-{\int }_1^{e}1dx\\ & =& e-\left(e-1\right)\\ & =& 1\end{array}$

Therefore, the area of the shaded region is $\boxed{1}$