Given a one-to-one mapping $u=u(x, y)$ and $v=v(x, y)$ between the region $D$ in the $(x, y)$-plane and the region $D^{\prime}$ in the $(u, v)$-plane, state the formula for transforming the integral $\iint_{D} f(x, y) d x d y$ into an integral over $D^{\prime}$, with the Jacobian expressed explicitly in terms of the partial derivatives of $u$ and $v$.

Let $D$ be the region $x^{2}+y^{2} \leqslant 1, y \geqslant 0$ and consider the change of variables $u=x+y$ and $v=x^{2}+y^{2}$. Sketch $D$, the curves of constant $u$ and the curves of constant $v$ in the $(x, y)$-plane. Find and sketch the image $D^{\prime}$ of $D$ in the $(u, v)$-plane.

Calculate $I=\iint_{D}(x+y) d x d y$ using this change of variables. Check your answer by calculating $I$ directly.