Answer
$112.09\ \ \mathrm{cm}^{3}/\mathrm{s}$
Work Step by Step
The flow rate is given by
$$
\begin{aligned}
2 \pi \int_{0}^{2} r v(r) d r &=2 \pi \int_{0}^{2} r\left(10-0.3 r-0.34 r^{2}\right) d r\\
&=2 \pi \int_{0}^{2}\left(10 r-0.3 r^{2}-0.34 r^{3}\right) d r \\
&=\left.2 \pi\left(5 r^{2}-0.1 r^{3}-0.085 r^{4}\right)\right|_{0} ^{2} \\
&=112.09\ \ \mathrm{cm}^{3}/\mathrm{s}
\end{aligned}
$$
