## Thomas' Calculus 13th Edition

(a) $\int_{e^{-x}}^1 \int_{0}^{\ln 3} f(x,y) dy dx$ (b) $\int_{-\ln y}^{\ln 3} \int_{1/3}^{1} f(x,y) dx dy$
(a) For vertical cross-sections, the region $R$ can be defined as: $R=$ { $( x,y) | e^{-x} \leq y \leq 1 , 0 \leq x \leq \ln (3)$} Hence, we have $\int_{e^{-x}}^1 \int_{0}^{\ln 3} f(x,y) dy dx$ (b) For horizontal cross-sections, the region $R$ can be defined as: $R=$ { $( x,y) | -\ln y \leq x \leq \ln 3 , \dfrac{1}{3} \leq y \leq 1$} Hence, we have $\int_{-\ln y}^{\ln 3} \int_{1/3}^{1} f(x,y) dx dy$