I = $\displaystyle \int$ between 0 to 1 dy, $\displaystyle \int$ 0 to y $\displaystyle y^3/(x^2 + y^2)$ dx
sketch the domain of integration
The outer integral is form x= 0 to x= 1. Draw the vertical lines, x= 0, y= 1. The inner integral is from y= 0 to $\displaystyle y= x^3$. Draw that line and curve. The region of integration is the region bounded by those.