I've been trying to solve this question in many ways, at first I tried a graphical approach, which, didn't bring me many results, not any correct ones at least… However, searching around, I found out about the "Heaviside function method", so this is my second attempt: $$ \int_{0}^{1}\int_{e^{y-1}}^{e^y} f(x,y)dxdy=\int_{0}^{1}\int_{1}^{e^y}f(x,y)dxdy+\int_{0}^{1}\int_{e^y-1}^{1}f(x,y)dxdy\\ =\int_{0}^{1}\int_{1}^{e}\Theta (e^y-x)f(x,y)dxdy+\int_{0}^{1}\int_{1/e}^{1}\Theta (x-e^{y-1})f(x,y)dxdy\\ =\int_{1}^{e}\int_{0}^{1}\Theta (y-\ln(x))f(x,y)dydx+\int_{1/e}^{1}\int_{0}^{1}\Theta (\ln(x)+1-y)f(x,y)dydx\\ =\int_{1}^{e}\int_{\ln(x)}^{1}f(x,y)dydx+\int_{1/e}^{1}\int_{0}^{\ln(x)+1}f(x,y)dydx $$ The first part(first integral) seems to be right, the second, however, does not, can anyone please help me find my mistake? Is it in the boundaries?
I assume having a hard time figuring out how to calculate and adjust the boundaries to the Heaviside function ""conversion"", due to the lack of information I found about this part. After reading this[link] I thought I had it figured out, but, after this result I'm not so sure anymore…