Double Integrals in Polar Coordinates — Question 10

Question 10

Let $D$ be the upper half-disk $x^2+y^2\le 9$, $y\ge 0$. Evaluate $$\iint_D(x^2+y^2+xy)dA.$$

Tasks

1. Convert the region and integrand to polar coordinates.

2. Use angular symmetry to eliminate the mixed term.

3. Evaluate the remaining integral and verify its sign.

Question 10 – Solution

Strategy. In polar form the radial term is $r^2$, while $xy=r^2\cos\theta\sin\theta$ cancels over $0\le\theta\le\pi$.

Step 1: Set up $$0\le\theta\le\pi,\quad 0\le r\le 3,$$ so $$I=\int_0^\pi\int_0^3r^3(1+\cos\theta\sin\theta)\,dr\,d\theta.$$

Step 2: Symmetry Since $$\int_0^\pi\cos\theta\sin\theta\,d\theta=0,$$ the $xy$ contribution vanishes, matching reflection symmetry across the $y$-axis.

Step 3: Evaluate $$I=\int_0^\pi\left[\frac{r^4}{4}\right]_0^3d\theta =\int_0^\pi\frac{81}{4}d\theta =\boxed{\frac{81\pi}{4}}.$$ The surviving integrand $x^2+y^2$ is nonnegative, so the positive result is necessary. Its scale is radius squared times half-disk area, as expected.