Answer
$e-1$
Work Step by Step
The domain $D$ for given region can be expressed as: $x \leq y \leq x+1$ and $0 \leq x \leq 1$
The iterated integral can be calculated as:
$\iint_{D} f(x,y) d A=\int_0^1 \int_{x}^{x+1} e^{x} dy dx\\=\int_0^1 (ye^x)_x^{x+1} dy \\=\int_0^1 [(x+1)e^x-xe^x] \ dx \\=\int_0^1 e^x \ dx \\=[e^x]_0^1 \\=e-1$
