Double Integral $\iint\limits_Se^{x+y}\,dx\,dy$ where $S=\{(x,y):|x|+|y|\le1\}$

Question

Let $S=\{(x,y):|x|+|y|\le1\}$. How to evaluate $$\iint_Se^{x+y}\,dx\,dy?$$ Please help. Thanks in advance.

Answer 1

It helps if you can visualize the set $S$. Notice that for any $x$ such that $\lvert x \rvert \leq 1$, we have $\lvert y \rvert \leq 1 - \lvert x \rvert$, or $$ -1 + \lvert x \rvert \leq y \leq 1 - \lvert x \rvert. $$ This can be broken down further, depending on the sign of $x$: $$ \begin{cases} -1+x \leq y \leq 1-x, & x \geq 0, \\ -1-x \leq y \leq 1+x, & x < 0. \end{cases} $$ Let's call these two triangular regions $S_+$ and $S_-$, respectively. Here's a picture of the region $S = S_+ \cup S_-$.

Integration will be with respect to $y$ first for a given fixed $x$ (satisfying $-1 \leq x \leq 1$), then with respect to $x$. Due to the symmetry of the region and the integrand, these can be interchanged. $$ \begin{align} \iint_S e^{x+y} \, dx \, dy &= \iint_{S_-} e^{x+y} \, dx \, dy + \iint_{S+} e^{x+y} \, dx \, dy \\ &= \int_{-1}^{0} \int_{-1-x}^{1+x} e^{x+y} \, dy \, dx + \int_{0}^{1} \int_{-1+x}^{1-x} e^{x+y} \, dy \, dx. \end{align} $$

Can you take it from there?