Answer
$\frac{8 \pi}{3} \ {\mathrm{cm}^{3}/\mathrm{s}}$
Work Step by Step
The flow rate is given by
$$
2 \pi \int_{1}^{3} r(0.5)(r-1)(3-r) d r\\
=\pi \int_{1}^{3}\left(-r^{3}+4 r^{2}-3 r\right) d r\\
=\left.\pi\left(-\frac{1}{4} r^{4}+\frac{4}{3} r^{3}-\frac{3}{2} r^{2}\right)\right|_{1} ^{3}\\
=\frac{8 \pi}{3} \ {\mathrm{cm}^{3}/\mathrm{s}}
$$
