$\int_{E} f$ where $f(x, y)=x^{2} y$ and $E$ is the region in $\mathbb{R}^{2}$ between the line $y=x-2$ and the parabola $x=4-y^{2}$

Question

• Evaulate the integral $\int_{E} f$ where $f(x, y)=x^{2} y$ and $E$ is the region in $\mathbb{R}^{2}$ between the line $y=x-2$ and the parabola $x=4-y^{2}$ Use Funini's thereom

My Attempt. The graph

Then we evaluate $\int_{-2}^{1} \int_{-2-y}^{4-y^2} {x^2y}$ $dx dy.$

Are the boundeds true? I think, it shoould be: $\int_{-2}^{1} \int_{2+y}^{4-y^2} {x^2y}$ $dx dy.$ May you check? Thanks...