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