Change of variables in triple integral Let $D$ be the region in $xyz-$space defined by inequalities $1 \le x \le 2, 0 \le xy \le 2 $ and $0\le z \le 1$. I want to evaluate $\displaystyle \int\int\int_D (x^2y + 3xyz) \text{dxdydz}$ by applying the following transformation in $uvw-$space:

$$u = x, v = xy, w =3z$$ 
over an appropriate region $G$.

So,  I sketch the region $D$ and write $x,y,z$ as 
$$x=u, \enspace y = \frac{v}{x} = \frac{v}{u} \enspace z = \frac{w}{3}$$ and found the Jacobian $J(u,v,w)$ as $\frac{1}{3u}$.

Now, I have two questions:

1.
 I know that I have to write the boundaries of $D$ in terms of $x,y,z$ which comes from the $6$-sides of it. However, since we have $0\le xy \le 2$ the bottom of $D$ is some kind of right triangle with $y=1$ line at the bottom, $x = 1$ line on the left side and some part of $y = \frac{1}{x}$ curve. So I have some kind of prism which has $5$-sides. In that case, how do we write $6$-conditions for the boundaries of $u,v$ and $w$?

2.For boundaries of $G$, I found $1\le u \le 2$ and $0 \le w \le 3$.  I found $u \le v \le u^2$ by using $0 \le y \le \frac{2}{x} $ and considering $1 \le u \le 2$. However, it is clear that these cannot be the boundaries.

I think the first question solves the second one. Thank you.
This is not a good example to apply change of variables.  As the limits of z do not depend on x or y and is constant between 0 and 1.  Only y depends on x.  Hence the integral can be written as

$$\int_{z=0}^{1}\int_{x=1}^{2}\int_{y=0}^{\frac{2}{x}}(x^2y + 3xyz)dydxdz = 2 + 3ln 2$$
This is not a good example to apply change of variables.  As the limits of z do not depend on x or y and is constant between 0 and 1.  Only y depends on x.  Hence the integral can be written as

$$\int_{z=0}^{1}\int_{x=1}^{2}\int_{y=0}^{\frac{2}{x}}(x^2y + 3xyz)dydxdz = 2 + 3ln 2$$