Find the volume of the solid S between the surfaces z equals 4 e Superscript x minus y and z equals negative 3 e Superscript x minus y​, where S intersects the​ xy-plane in the region Upper R equals StartSet (x comma y ): 0 less than or equals x less than or equals y comma 0 less than or equals y less than or equals 1 EndSet.

Answer:[tex]V=\frac{7}{e}[/tex]Step-by-step explanation:[tex]z_{1} =4e^{x-y}[/tex][tex]z_{2} =-3e^{x-y}[/tex][tex]0\leq x\leq y[/tex][tex]0\leq y\leq 1[/tex][tex]V=\int\limits^._. {\int\limits^._. [z_{1}(x,y)-z_{2}(x,y)]\, dx } \, dy[/tex][tex]V=\int\limits^1_0 {\int\limits^y_0 [4e^{x-y}-(-3e^{x-y} )]\, dx } \, dy[/tex][tex]V=\int\limits^1_0 {\int\limits^y_0 [4e^{x-y}+3e^{x-y}]\, dx } \, dy[/tex][tex]V=\int\limits^1_0 {\int\limits^y_0 [7e^{x-y}]\, dx } \, dy[/tex][tex]V=\int\limits^1_0 {7e^{-y}(e^{x})^{y}_{0}} \, dy[/tex][tex]V=\int\limits^1_0 {7e^{-y}(e^{y}-e^{0})} \, dy[/tex][tex]V=\int\limits^1_0 {[7-7e^{-y}]} \, dy[/tex][tex]V=(7y+7e^{-y} )^{1} _{0}[/tex][tex]V=(7(1)+7e^{-1} )-(7(0)+7e^{0} )[/tex][tex]V=7+7e^{-1} -0-7[/tex][tex]V=7e^{-1} [/tex][tex]V=\frac{7}{e}[/tex]Hope this helps!