What is the region $\displaystyle R $ over which you integrate when evaluating the double integral $\displaystyle \int_{1}^{2} \int_{1}^{x} \frac{x}{\sqrt{x^2+y^2}} \ dy \ dx $? Rewrite this as an iterated integral first with respect to $\displaystyle x $, then with respect to $\displaystyle y $. Evaluate this integral. Which order of integration is easier?

Is the region simply $\displaystyle R = \{(x,y) \ | \ 1 \leq x \leq 2, \ 1 \leq y \leq x \} $?

Rewriting it, I get $\displaystyle \int_{1}^{x} \int_{1}^{2} \frac{x}{\sqrt{x^2+y^2}} \ dx \ dy $. And evaluating it should be straightforward (guessing that it will be easier to do it first with respect to $\displaystyle x $).