Volume of Generalized Tetrahedron in R n .I'm having difficulty finding the volume of a...

Step 1
Your approach is fine, but the allowable range in $x_3$ is $1-x_1-x_2$, so that should be your integrand. It is probably easier to define the n-volume of a k sided simplex as $V_n(k)$ and recognize that $V_n(k)={\int }_0^k{V}_{n-1}(x)dx$.
Step 2
Now each integral is a single one. If you do the first few, you will see a pattern emerge, which you can prove by induction.

Step 1
It is more accurate to write for two dimensions like this:
${\int }_0^{1}d{x}_1{\int }_0^{x_1}d{x}_2$
Step 2
For third dimension it will be:
${\int }_0^{1}d{x}_1{\int }_0^{x_1}d{x}_2{\int }_0^{x_2}d{x}_3$
So you can easily write the expression for higher dimensions. My advise is to write the domain bound by hyperplanes more carefully. Your way is too difficult for further integration I guess.