How to calculate $\int_{P}\sqrt{x+y}d\lambda ^{2}(x,y)$

Question

Let be $$P=\left \{ (x,y)\in\mathbb{R}^{2}\mid 0<x+y<1,0<2x-3y<4 \right \}$$ I wonder how to calculate $$\int_{P}\sqrt{x+y}d\lambda ^{2}(x,y)$$ using $$\begin{pmatrix} x\\ y \end{pmatrix} \mapsto \begin{pmatrix} 1 & 1\\2 &-3 \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}$$