Calculating the integral in measure theory I'm very new to measure theory, so I don't really understand how to do this:
$\int_Axe^yd\lambda^2$ for A:=$\left\{(x,y)\in R : 0\le x\le 1, 0\le y\le \frac1 2 x^2\right\}$
Do I calculate it using regular integral rules? I'm sorry if this sounds like a I don't know anything question, but I am truly lost and I hope that someone can still help out, thanks in advance!

Assuming $\lambda^2$ is supposed to be the Lebesgue measure on $\mathbb R^2$, you can simply treat this integral as a Riemann integral, since $(x,y)\mapsto x\cdot e^y$ is continuous and therefore Riemann integrable. In this case, the Riemann integral and Lebesgue integral are equal, see here.
Now, the Riemann integral can be written as a double integral of the form $$\int_
*^*\int_*^*xe^y dxdy$$ You'll just have to figure out the integration boundaries $(*)$.
