Answer
$\dfrac{17 \sqrt {17}-1}{4}$
Work Step by Step
Consider $\vec{r} (x,z) =x i+x^2 j+zk$
So, $\vec{r_x} \times \vec{r_z}=2x i-j +0 k$
and $|\vec{r_x} \times \vec{r_z}| =\sqrt {4x^2+1}$
Now, $\int_{0}^2 \int_{0}^3 6 (x,y,z) \ d \theta =\int_{0}^2 \int_{0}^3 x \sqrt {4x^2+1} \ dz \ dx=\int_0^2 (3x \sqrt {4x^2+1} ) \ dx$
Consider $a =4x^2+1$ and $da=8x dx$
or, $=\dfrac{3}{8} \int_{1}^{17} \sqrt a da$
or, $=\dfrac{3}{8} [\dfrac{2}{3} a^{3/2}]_{1}^{17}$
or, $=\dfrac{17 \sqrt {17}-1}{4}$
