Q:

How to calculate the volume of an $N$-dimensional hyper rectangular

Let $H$ be an $N$-dimensional rectangular with edges of length $L_i$, $i=1,2,\dots,N$. Is the volume simply the product of the $L_i$, $i=1,2,\dots,N$?
$$ \text{Volume} = L_1 \times L_2 \times \dots \times L_N$$

A:

This is an easy one. Short response: Yes
"Long" response: The volume of a region is just the integral of the function $f(x_1,x_2,x_3...)=1$ over the region. In the case of a n-rectangle
$$V=\int_R\text{d}x_1\text{d}x_2\dots=\int_{L_1}\text{d}x_1\int_{L_2}\text{d}x_2...=L_1\cdot L_2\cdot...$$

