Double integration with $e^{-x^2}$

Question

I am learning Fubini right now and I want to integrate $$ \int_U e^{-x^2}y d\lambda_2 , $$ whereby $$ U=\left\{(x, y) \in \mathbb{R}^{2}: 0 \leq y \leq 1, \quad y^{2} \leq x \leq 1\right\} $$

But as you may already see, we cannot do this without using the error function so I thought that one can maybe use polar coordinates and then apply fubini but I dont know how the set $U$ changes and what the new limits are.

Answer 1

Thanks to Geoffrey Trang I got

$$ \int_0^1 \int_0^{\sqrt{x}} e^{-x^2}y dy dx = \int_0^1 \frac{xe^{-x^2}}{2}dx = (e-1)/4e $$