$(1)$ \[ \begin{align} Q&=\int_AudA\\ &=\int_o^Ru2\pi rdr\\ &=\int_o^RU_{max}(\frac{Rr}{R})^{1/n}2\pi rdr\\ &=\frac{2\pi U_{max}}{R^{\frac{1}{n}}}\int_o^Rr(R-r)^{1/n}dr \end{align}\] \[\begin{align}\int_o^Rr(R-r)^{1/n}dr&=\left[r\cdot (-1)\frac{1}{\frac{1}{n}+1}(R-r)^{\frac{1}{n}+1}\right]_o^R\\ &-\int_o^R(-1)\frac{1}{\frac{1}{n}+1}(R-r)^{\frac{1}{n}+1}dr\\ &=-\frac{n}{n+1}\left[r(R\cdot r)^{\frac{n+1}{n}}\right]_o^R+\int_o^R\frac{n}{n+1}(R-r)^{\frac{n+1}{n}}dr\\ &=\frac{n}{n+1}\left[\frac{(-1)}{\frac{n+1}{n}+1}(R-r)^{\frac{n+1}{n}+1}\right]_o^R\\ &=\frac{n}{n+1}\frac{n}{2n+1}\left[- (R-r)^{\frac{2n+1}{n}}\right]_o^R\\ &=\frac{n^2}{(n+1)(2n+1)}R^{\frac{2n+1}{n}}\end{align}\] \[\begin{align} Q&=\frac{2\pi U_{max}}{R^{1/n}}\times\frac{n^2}{(n+1)(2n+1)}R^{\frac{2n+1}{n}}\\ &=2\pi u_{max}R^2\frac{n^2}{(n+1)(2n+1)} \end{align}\] $(2)$ \[ \begin{align} \upsilon=\frac{Q}{A}&=\frac{1}{\pi R^2}2\pi U_{max}R^2\frac{n^2}{(n+1)(2n+1)}\\ &=\frac{2U_{max}n^2}{(n+1)(2n+1)} \end{align}\]