Question. Evaluate the double integral $$ \iint_{D} e^{(x+y)^{2}} \, dxdy $$ over the region $D=\left \{ \left ( x,y \right ):0\leq x-y\leq x+y\leq 1 \right \}$.
I can get the region which is bounded by $x+y=1$, $y=x$, and $x>0$. But I can't figure out how to integrate it. Please help me.
Hint: use the substitution $$\begin{cases}u:=x+y\\v:=x-y\end{cases}$$ Then we get $$\iint_D e^{(x+y)^2}~dxdy=\iint_{C} e^{u^2}\left|\frac{\partial(x,y)}{\partial (u,v)}\right|~dudv$$ where $C:=\{(u,v):0\leq v\leq u\leq 1\}$ and $$\left|\frac{\partial(x,y)}{\partial (u,v)}\right|=\det\begin{bmatrix}\frac{\partial x}{\partial u}&\frac{\partial x}{\partial v}\\\frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}\end{bmatrix}$$ Notice that the new integral is independent of $v$ (determine the bounds on this new integral and this will be very nice!).