Answer
$128 \pi \
\ \mathrm{cm}^{3}/\mathrm{s}$
Work Step by Step
The flow rate is given by
$$
2 \pi \int_{0}^{R} r v(r) d r=2 \pi \int_{0}^{4} r\left(16-r^{2}\right) d r\\
=\left.2 \pi\left(8 r^{2}-\frac{1}{4} r^{4}\right)\right|_{0} ^{4}=128 \pi \
\ \mathrm{cm}^{3}/\mathrm{s}
$$
