Consider $y=\ln x$ in the interval $[1,e]$. 

Obviously, the area between $f(x)$ and the $x$-axis in terms of single-variable calculus is $\int\_1^e \ln x~dx$.  

However, an equivalent evaluation could be considered in the form of a double integral, namely in the form of $\int \int\_D f ~dA$, where $D$ represents the domain of integration, and $A$ the (vector) variable to be integrated by. 

I understand that the domain of the integral should be $\int\_1^e \int\_0^{\ln x}$ quite clearly, but why is $f$ in this case $1$?
(i.e. Why does $\int\_1^e \ln x~dx \equiv \int\_1^e \int\_0^{\ln x} 1 ~dy~dx$ hold true?)