In the double integral double integral over D z(x,y) dx
In the double integral $\iint_D z(x,y)\,dx\,dy$, convert to polar coordinates $\varphi$ and $r$ and write the limits of integration for both possible orders. The region $D$ is bounded by the curve $(x^2+y^2)^2=a^2(x^2-y^2)$ where $x>0$.
Explanation of Method
Substituting $x=r\cos\varphi$, $y=r\sin\varphi$ turns $x^2-y^2$ into $r^2\cos2\varphi$, collapsing the quartic curve into the compact polar form $r=a\sqrt{\cos2\varphi}$ — the classic lemniscate equation. The requirement $x>0$ combined with $\cos2\varphi\geqslant0$ pins down the angular range to one loop.
- Substitute polar coordinates to derive $r=a\sqrt{\cos2\varphi}$.
- Determine the $\varphi$-range from $\cos2\varphi\geqslant0$ and $x>0$.
- Write the direct order.
- Invert for the reversed order.
Step 1 · Convert to polar coordinates
$x^2+y^2=r^2$ and $x^2-y^2=r^2\cos2\varphi$, so $(x^2+y^2)^2=a^2(x^2-y^2)$ becomes $r^4=a^2r^2\cos2\varphi$, and for $r\neq0$:
$$ r=a\sqrt{\cos2\varphi}. $$
Step 2 · Determine the range of $\varphi$
Real $r$ requires $\cos2\varphi\geqslant0$, i.e. $-\frac\pi4\leqslant\varphi\leqslant\frac\pi4$ (mod $\pi$). Combined with $x>0$ ($\cos\varphi>0$, i.e. $-\frac\pi2<\varphi<\frac\pi2$), the relevant loop is:
$$ -\frac{\pi}{4}\leqslant\varphi\leqslant\frac{\pi}{4}. $$
Step 3 · Describe the region
$D$ is bounded outward by $r=a\sqrt{\cos2\varphi}$ and inward by the origin, for $-\frac\pi4\leqslant\varphi\leqslant\frac\pi4$.
Step 4 · Order 1 — integrate $r$ first, then $\varphi$
$$ \iint_D z\,dx\,dy = \int_{-\pi/4}^{\pi/4} d\varphi \int_0^{a\sqrt{\cos2\varphi}} z(r\cos\varphi,r\sin\varphi)\,r\,dr. $$
Step 5 · Order 2 — integrate $\varphi$ first, then $r$
From $r=a\sqrt{\cos2\varphi}$: $\cos2\varphi=\frac{r^2}{a^2}$, so $\varphi=\pm\frac12\arccos\!\left(\frac{r^2}{a^2}\right)$. The maximum $r$ is $a$ (at $\varphi=0$), so $0\leqslant r\leqslant a$:
$$ \iint_D z\,dx\,dy = \int_0^{a} dr \int_{-\frac12\arccos(r^2/a^2)}^{\frac12\arccos(r^2/a^2)} z(r\cos\varphi,r\sin\varphi)\,r\,d\varphi. $$
Additional Notes
- The lemniscate is one of the classic curves whose Cartesian equation looks intimidating but simplifies beautifully in polar form.
- The condition $x>0$ selects the right-hand loop of the two-loop lemniscate; the left loop would use $\frac{3\pi}{4}\leqslant\varphi\leqslant\frac{5\pi}{4}$.
Result
The limits of integration are: