Find the Area $A$ 
My attempt:

$A=\ln {e^2}-\ln 1=2$

Is this a correct answer?

Yes, your answer is correct. The area asked is an elementary application of integration. Specifically : 
$$A = \int_1^{e^2}\frac{1}{x}dx=\big[ \ln(x)\big]_1^{e^2}= \ln(e^2)-\ln(1)=2\ln(e)=2$$
