Answer
$$V = 9$$
Work Step by Step
$$\eqalign{
& V = \int_1^2 {\int_0^1 {\left( {12 - {x^2} - 2{y^2}} \right)} } dydx \cr
& {\text{Integrating with respect to }}y \cr
& V = \int_1^2 {\left[ {12y - {x^2}y - \frac{2}{3}{y^3}} \right]} _0^1dx \cr
& {\text{Evaluate the limits}} \cr
& V = \int_1^2 {\left[ {12\left( 1 \right) - {x^2}\left( 1 \right) - \frac{2}{3}{{\left( 1 \right)}^3}} \right]} dx \cr
& V = \int_1^2 {\left( {\frac{{34}}{3} - {x^2}} \right)} dx \cr
& {\text{Integrating}} \cr
& V = \left[ {\frac{{34}}{3}x - \frac{1}{3}{x^3}} \right]_1^2 \cr
& V = \left[ {\frac{{34}}{3}\left( 2 \right) - \frac{1}{3}{{\left( 2 \right)}^3}} \right] - \left[ {\frac{{34}}{3}\left( 1 \right) - \frac{1}{3}{{\left( 1 \right)}^3}} \right] \cr
& V = \frac{{60}}{3} - 11 \cr
& V = \frac{{27}}{3}=9 \cr} $$
