Answer
$$1$$
Work Step by Step
$$Area=\int_{\ln 6}^{\ln 7} \int_{0}^{\ln 2} \int_{\ln 4}^{\\ln 5} e^{(x+y+z)} \ dx \ dy \ dz \\=\int_{\ln 6}^{\ln 7} \int_{0}^{\ln 2} [e^{(\ln 5+y+z)}-e^{(\ln 4 +y+z)} ] \ dy \ dz \\=\int_{\ln 6}^{\ln 7} \int_{0}^{\ln 2} e^{y+z} \ dy \ dz \\=\int_{\ln 6}^{\ln 7} [ e^{y+z}]_{0}^{\ln 2}\ dz \\=\int_{\ln 6}^{\ln 7} 2e^{z} dz -\int_{\ln 6}^{\ln 7} e^z \ dz \\=\int_{\ln 6}^{\ln 7} e^{z} \ dz \\=e^{\ln 7}-e^{\ln 6} \\=1$$
