Answer
The number of deer is $\approx 61 \ \mathrm{deer}$.
Work Step by Step
The number of deer is given as follows
$$
2 \pi \int_{2}^{5} r(150)\left(r^{2}+2\right)^{-2} d r=-\left.150 \pi\left(\frac{1}{r^{2}+2}\right)\right|_{2} ^{5}\\=-150 \pi\left(\frac{1}{27}-\frac{1}{6}\right) \approx 61 \ \mathrm{deer}
$$
