Volume is calculate as $\iint \limits_V f(x,y) \,dy \,dx$
so, $\int^{12}_0\int^x_0 z(x,y) \,dy \,dx$
$=\int^{12}_0\int^x_0 (x+y) \,dy \,dx$
$=\int^{12}_0 (xy+\frac{y^2}{2}) |^x_0 \,dx$
$=\int^{12}_0 (x^2+\frac{x^2}{2}) \,dx$
$=\frac{3}{2}\int^{12}_0 x^2 \,dx$
$=\frac{3}{2}\left ( \frac{x^3}{3} |^{12}_0 \right )$
$=864$