Assuming that phi and r are polar coordinates, change
Assuming that $\varphi$ and $r$ are polar coordinates, change the order of integration: $$ \int_{-\pi/2}^{\pi/2} d\varphi \int_0^{\cos\varphi} z(\varphi,r)\,dr. $$
Explanation of Method
This is the same style of order-reversal as in Cartesian coordinates, just with $(\varphi,r)$ in place of $(x,y)$. We find the maximum $r$ over the given $\varphi$-range, then invert $r=\cos\varphi$ to find the $\varphi$-bounds for each fixed $r$.
- Find the maximum value of $r=\cos\varphi$ on $[-\pi/2,\pi/2]$.
- Invert $r=\cos\varphi$ to bound $\varphi$ for fixed $r$.
- Verify the reversed integral at the endpoints.
Step 1 · Identify the region
From $\int_{-\pi/2}^{\pi/2} d\varphi\int_0^{\cos\varphi} z\,dr$: $-\frac\pi2\leqslant\varphi\leqslant\frac\pi2$, $0\leqslant r\leqslant\cos\varphi$ (valid since $\cos\varphi\geqslant0$ throughout this range).
Step 2 · Determine the range of $r$
The maximum of $\cos\varphi$ on $[-\pi/2,\pi/2]$ is $\cos(0)=1$, so $0\leqslant r\leqslant1$.
Step 3 · Find bounds on $\varphi$ for fixed $r$
We need $\cos\varphi\geqslant r$. Since $\cos\varphi$ decreases from $1$ (at $\varphi=0$) out to $0$ (at $\varphi=\pm\pi/2$), the condition $\cos\varphi\geqslant r$ holds on a symmetric interval around $0$: $\varphi=\pm\arccos(r)$ marks the boundary, so
$$ -\arccos(r) \leqslant \varphi \leqslant \arccos(r). $$
Step 4 · Write the integral with the reversed order
$$ \int_0^1 dr \int_{-\arccos(r)}^{\arccos(r)} z(\varphi,r)\,d\varphi. $$
Step 5 · Verification
At $r=0$: $\varphi$ ranges from $-\arccos(0)=-\pi/2$ to $\arccos(0)=\pi/2$. ✓ At $r=1$: $\varphi$ ranges from $-\arccos(1)=0$ to $\arccos(1)=0$ (a single point). ✓ At $r=\frac12$: $\varphi$ ranges from $-\pi/3$ to $\pi/3$. ✓
Additional Notes
- The underlying region is the disk $r\leqslant\cos\varphi$, which (converted to Cartesian) is a circle of radius $\frac12$ centered at $\left(\frac12,0\right)$ — the same region as Problem 11, Part $2^\circ$, viewed in polar form.
Result
The order-reversed integral is: