Reversing the order of integration to solve the double integral

I am trying to solve the double integral:

$ \int_{0}^1\int_{1-x}^{\sqrt(1-x)}$ $ e^{{y^2/2}-{y^3/3}}$ $ dydx$

by reversing the order of integration, however, I am unsure how to go about doing it. Is it right to say that initially:

$ \sqrt(1-x)$ $ ≤y≤(1-x)$ and $ 0≤x≤1$ . After reversing the order, we get $ 1-y^2≤x≤1-y$ and $ 0≤y≤1$ , hence the reversed order of integration will be:

$ \int_{0}^1\int_{1-y}^{1-y^2}$ $ e^{{y^2/2}-{y^3/3}}$ $ dxdy$ ?