v = \(\int\limits_{\theta=0}^{\theta=\pi/2}\)(\(\int\limits_{r=0}^{r=1}\)(\(\int\limits_{z=0}^{z=\sqrt{4-r^2}}rdz )dr\))
= \(\int\limits_0^{\pi/2}\)(\(\int\limits_0^1r\sqrt{4-r^2}dr\)) dθ
(By taking 4 - r2 = t2, ⇒ -2rdr = 2tdt ⇒ rdr = -tdt)
= \(\int\limits_0^{\pi/2}(\int\limits_{\sqrt3}^2t^2dt)d\theta\)
= \(\int\limits_0^{\pi/2}[\frac{t^3}3]^2_{\sqrt3}d\theta\)
= \(\frac{\pi}2(\frac83-\frac{3\sqrt3}3)\) cubic units
= \(\frac{\pi}2(\frac83-\sqrt3)\) cubic units
