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 a triangle with vertices $O(0,0)$, $A(1,0)$, $B(1,1)$.
Explanation of Method
We first identify the equations of the three sides, then describe $D$ as a vertically simple region (fixed $x$, $y$ between two boundary curves) and as a horizontally simple region (fixed $y$, $x$ between two boundary curves).
- Find the equations of sides $OA$, $AB$, $OB$.
- Describe $D$ for vertical strips ($dy\,dx$).
- Describe $D$ for horizontal strips ($dx\,dy$).
- Check both orders give the same area.
Step 1 · Sketch the region
The triangle has vertices $O=(0,0)$, $A=(1,0)$, $B=(1,1)$ — a right triangle in the first quadrant.
Step 2 · Find the equations of the sides
Side $OA$ (from $(0,0)$ to $(1,0)$): $y=0$, $0\leqslant x\leqslant1$.
Side $AB$ (from $(1,0)$ to $(1,1)$): $x=1$, $0\leqslant y\leqslant1$.
Side $OB$ (from $(0,0)$ to $(1,1)$): slope $1$, so $y=x$, $0\leqslant x\leqslant1$.
Step 3 · Describe the region
$D$ is bounded below by $y=0$, on the right by $x=1$, and above/left by $y=x$.
Step 4 · Order 1 — integrate $y$ first, then $x$
For fixed $x\in[0,1]$, $y$ runs from the bottom $y=0$ to the diagonal $y=x$:
$$ \iint_D z(x,y)\,dx\,dy = \int_0^1\int_0^x z(x,y)\,dy\,dx. $$
Step 5 · Order 2 — integrate $x$ first, then $y$
For fixed $y\in[0,1]$, $x$ runs from the diagonal $x=y$ to the right edge $x=1$:
$$ \iint_D z(x,y)\,dx\,dy = \int_0^1\int_y^1 z(x,y)\,dx\,dy. $$
Step 6 · Verification — check areas
With $z(x,y)=1$: Order 1 gives $\int_0^1 x\,dx=\frac12$; Order 2 gives $\int_0^1(1-y)\,dy=\frac12$. Both give area $\frac12$, correct for a right triangle with legs $1$. ✓
Summary Table
| Order | Outer limits | Inner limits |
|---|---|---|
| $dy\,dx$ | $0\leqslant x\leqslant1$ | $0\leqslant y\leqslant x$ |
| $dx\,dy$ | $0\leqslant y\leqslant1$ | $y\leqslant x\leqslant1$ |
Additional Notes
- The triangle is bounded by $y=0$ (bottom), $x=1$ (right), and $y=x$ (hypotenuse).
- Both descriptions cover the same region and give the same area $\frac12$.
Result
The two equivalent forms are: