Section 15.3: Integration in Polar Coordinates

SOLUTION The region \(R\) can be described as \[ R = \{(x, y) | y \ge 0, 1 \le x^2 + y^2 \le 4\} \] It is the half-ring shown in Figure 1(b), and in polar coordinates it is given by \(1 \le r \le 2, 0 \le \theta \le \pi\). Therefore, by Formula 2, \[ \iint_R (3x + 4y^2) dA = \int_0^\pi \int_1^2 (3r \cos \theta + 4r^2 \sin^2 \theta) r dr d\theta \] \[ = \int_0^\pi \int_1^2 (3r^2 \cos \theta + 4r^3 \sin^2 \theta) dr d\theta \]

\[ = \int_0^{\pi} \left[r^3 \cos \theta + r^4 \sin^2 \theta \right]_{r=1}^{r=2} d\theta = \int_0^\pi (7 \cos \theta + 15 \sin^2 \theta) d\theta \] \[ = \int_0^\pi \left[ 7 \cos \theta + \frac{15}{2}(1 - \cos 2\theta) \right] d\theta \] \[ = \left[ 7 \sin \theta + \frac{15\theta}{2} - \frac{15}{4}\sin 2\theta \right]_0^\pi = \frac{15\pi}{2} \]