Physics flow rate

Question

Find the flow rate through a tube of radius $5~\mathrm{cm}$, assuming that the velocity of fluid particles at a distance $r~\mathrm{cm}$ from the center is $v(r) = 49-r^2~\mathrm{cm/s}$.

Answer 1

Using polar coordinates:

$$\Phi=\iint_{C}v(r)dS$$ where $C$ is a circle centered at the origin and radius $r=5$, then \begin{align} \Phi&=\int_{\theta=0}^{2\pi}\int_{r=0}^5(49-r^2)r\,drd\theta \end{align}