Answer
$\int_0^2 \int_{1}^{e^x} f(x,y) dy dx$
and $\int_{1}^{e^2} \int_{\ln (y)}^{2} f(x,y) dx dy$
Work Step by Step
(a) For vertical cross-sections, the region $R$ can be defined as:
$R=$ { $( x,y) | 1 \leq y \leq e^x , 0 \leq x \leq 2$}
Hence, we have $\int_0^2 \int_{1}^{e^x} f(x,y) dy dx$
(b) For horizontal cross-sections, the region $R$ can be defined as:
$R=$ { $( x,y) | \ln y \leq x \leq 2 , 1 \leq y \leq e^2$}
Hence, we have $\int_{1}^{e^2} \int_{\ln (y)}^{2} f(x,y) dx dy$
