I have the following set:
$$M=\left\{ (x,y,z) \ | \ 0 \leq x \leq 1, 0 \leq y \leq x, 0 \leq z\leq xy \right\}$$
I am completely new to multidimensional analysis, so how do I even start when there is no $f(x,y)$ given here? Here is my attempt:
$$\int_0^{xy} \int_0^x \int_0^1 1 \ dx \ dy \ dz$$
On
Expanding a bit on my functional determinant comment. If x is measured in decimeters, y in centimeters and z in meters your functional matrix would be
$$\begin{bmatrix} 0.1&0&0\\0&0.01&0\\0&0&1\end{bmatrix}$$ and the determinant would in this simple example be diagonal product $0.1\cdot 0.01 \cdot 1 = 0.001$. So the functional determinant keeps track of how the volume is "rescaled" at each point in your set. In our case each volume element would be 1 thousandth time as big as if we measured each dimension in meters.
If all dimensions are unit $1$ lengths, we would get the functional determinant of the identity matrix, which is where you get your $1$ in your integral from.
Make that $$\int_0^1\int_0^x\int_0^{xy} 1\,dz\,dy\,dx.$$