Write the limits of integration for the double integral
Write the limits of integration for the double integral $\iint_D z(x,y)\,dx\,dy$ for both possible orders of integration. The region $D$ is defined by the inequality $|x|+|y|<1$.
Explanation of Method
The diamond $|x|+|y|\leqslant1$ has a different linear equation for its boundary in each quadrant. Splitting into $x\geqslant0$ and $x\leqslant0$ (or $y\geqslant0$ and $y\leqslant0$) lets us combine the quadrant equations into a single compact bound using $|x|$ or $|y|$.
- Write the boundary equation in each quadrant.
- Combine quadrant pairs into piecewise bounds for each order.
- Simplify using $|x|$ and $|y|$ notation.
Step 1 · Identify the region
$|x|+|y|\leqslant1$ is a diamond (square rotated $45^\circ$) centered at the origin with vertices $(\pm1,0)$ and $(0,\pm1)$.
Step 2 · Express the boundary in each quadrant
First quadrant ($x,y\geqslant0$): $x+y=1$. Second quadrant ($x\leqslant0,y\geqslant0$): $y=1+x$. Third quadrant ($x,y\leqslant0$): $y=-1-x$. Fourth quadrant ($x\geqslant0,y\leqslant0$): $y=x-1$.
Step 3 · Order 1 — integrate $y$ first, then $x$
For $-1\leqslant x\leqslant0$: upper boundary $y=1+x$, lower boundary $y=-1-x$. For $0\leqslant x\leqslant1$: upper boundary $y=1-x$, lower boundary $y=x-1$.
$$ \iint_D z\,dx\,dy = \int_{-1}^{0}\int_{-1-x}^{1+x} z\,dy\,dx + \int_{0}^{1}\int_{x-1}^{1-x} z\,dy\,dx = \int_{-1}^{1}\int_{-(1-|x|)}^{1-|x|} z\,dy\,dx. $$
Step 4 · Order 2 — integrate $x$ first, then $y$
By the symmetric argument with $x$ and $y$ swapped:
$$ \iint_D z\,dx\,dy = \int_{-1}^{0}\int_{-1-y}^{1+y} z\,dx\,dy + \int_{0}^{1}\int_{y-1}^{1-y} z\,dx\,dy = \int_{-1}^{1}\int_{-(1-|y|)}^{1-|y|} z\,dx\,dy. $$
Summary Table
| Order | Compact form |
|---|---|
| $dy\,dx$ | $\int_{-1}^{1}\int_{-(1-|x|)}^{1-|x|} z\,dy\,dx$ |
| $dx\,dy$ | $\int_{-1}^{1}\int_{-(1-|y|)}^{1-|y|} z\,dx\,dy$ |
Additional Notes
- Writing the bounds as $\pm(1-|x|)$ (or $\pm(1-|y|)$) avoids splitting into two separate integrals when $z$ itself does not require it.
- The diamond is symmetric under $x\to-x$ and $y\to-y$, which both compact forms reflect.
Result
The two equivalent forms are: