\begin{align} \int_{-1}^1 dx \int_0^{1-x^2} dy \int_{x^2y+y^2}^y dz &= \int_{-1}^1 dx \int_0^{1-x^2} dy \left( -y^2 + \left( 1-x^2 \right) y \right) \\ &= \int_{-1}^1 dx \left[ - \frac{y^3}{3} + \left( 1-x^2 \right) \frac{y^2}{2} \right]_0^{1-x^2} \\ &= \frac{1}{6} \int_{-1}^1 dx \left( 1 - x^2 \right)^3 \\ &= \frac{1}{3} \int_0^1 dx \left( 1 - 3x^2 + 3x^4 - x^6 \right) \\ &= \frac{16}{105} \end{align}