Change of variables on integral of multivariate gaussian

Question

I have a question on the change of variables for multivariate Gaussian integrals.

Suppose that we have two independent random variables $X_1$ and $X_2$ where $X_i \sim N(0, \sigma_i^2)$ for $i=1,2$. Consider the linear transformation $\mathbf{y} = T(\mathbf{x}) = (2x_1 + x_2, x_1 + 10x_2)$, I want to find the integral

$$ V = \int_R f_y(\mathbf{y}) d\mathbf{y}, $$ where $f_y(\mathbf{x})$ is the joint pdf of $\mathbf{y}$ and $R$ is the region where both coordinates are positive.

My questions are as follows:

1. To calculate $V$, can I just apply the trick from the change of variables and claim that write $V$ as a function of $|\text{det}(T)|$ and the following integral $W$? $$ W = \int_R f_x(\mathbf{x}) d\mathbf{x} $$

$W$ is easier to evaluate than $V$ because the two variables are independent. $|\text{det} T|$ is easy to compute and it is independent of $\mathbf{x}$ because $T$ is a linear transformation. I wanted to claim that $V = W |\text{det}(T)|$ by interpreting $V$ and $W$ are areas, but this is not quite right and I am not sure what I am missing.

2. If question 1 is something that works, does the same idea work for normal that are having means being nonzero as well?

Thank you!