Evaluate the Double Integral at the Following Region

Question

Problem Statement

Evaluate $\iint_R xdA$, where R is the region bounded by $x=\ln(y)$, $x=0$,$y=e$.

My Work:

$$\int_0^e\int_0^{\ln(y)}x\ dx\ dy$$
or
$$\int_0^1 \int_{e^x}^e x\ dy\ dx$$

My Problem

Which integral setup is correct I tried doing the problems both ways Type 1, and Type 2, and I got different answers for both? I am confused generally on how to setup a Type 1, and Type 2 double integral for the same problem. Thank you for your time.

Answer 1

The type 2 integral is not properly set. It should be

$$ \int_1^e \int_0^{\ln y}x \, dx\, dy. $$