Answer
$2$
Work Step by Step
The area is given by $A=\int_0^{2} (x-1) e^x \ dx$
We will solve the given integral by using integrate-by-parts formula such as: $\int udv=uv-\int v du$
Here, $u=x-1$ and $dv=e^x dx \implies v=e^x$
$\displaystyle \int_0^{2} (x-1) e^x \ dx =(x-1) e^x-\int_0^2 e^x dx$
or, $=[(x-1) e^x -e^x]_0^2$
or, $=[(2-1) e^2 -e^2]-[(0-1) e^0 -e^0]$
or, $=0+2$
Therefore, the required area is: $Area=2$
