
$V = \pi {\displaystyle {\int_{y_1}}^{y_2}} ({x_R}^2 - {x_L}^2) \, dy$
$V = \pi {\displaystyle {\int_0}^\sqrt{2}} [ \, 2^2 - (y^2)^2 \, ] \, dy$
$V = \pi {\displaystyle {\int_0}^\sqrt{2}} [ \, 4 - y^4 \, ] \, dy$
$V = \pi \left[ 4y - \dfrac{y^5}{5} \right]_0^\sqrt{2}$
$V = \pi \left[ 4(\sqrt{2}) - \dfrac{(\sqrt{2})^5}{5} \right]$
$V = 14.22 \, \text{ unit}^3$ answer