I've got the following:

\begin{align}
\int\_{0}^{1}\int\_{0}^{y^{2}}\frac{y}{x^{2}+y^{2}}\ dx\ dy&=\int\_{0}^{1}\left.\arctan{\left(\frac{x}{y}\right)}\right|\_{x=0}^{x=y^{2}}\ dy\\
&=\int\_{0}^{1}\arctan{(y)}\ dy\\
&=y\arctan{(y)}-\int\_{0}^{1}\frac{y}{1+y^{2}}\ dy
\end{align}

I can figure out the integral, but my question is - what should be done about the values of y on the left? I've never done IBP before on a definite integral, so it never came up.

This is supposed to be done without switching the order of integration - I had considered that route but that's not until my next assignment.