A question about multiple integral

Question

How to compute the multiple integral

$$\int \int...\int_{(D)} dx_1dx_2...dx_n, \ \ D:-1\le x_1,x_2,\ldots ,x_n\le1, -1\le x_1+x_2+\cdots +x_n\le1.$$

Thanks in advanced for your help!

Answer 1

As @tired mentioned, you could use 2D and 3D to visualize it.

We can think about it as a n-D cube with side length $2$. And this cube is divided into 3 parts by the two planes $x_1+...+x_n=\pm 1$.

Consider the distance from the farthest point $(1,...,1)$ to the plane $x_1+...x_n=1$ (the other side is similar). The distance is $\frac{n-1}{\sqrt{n}}$. The distance from the origin to the plane is $\frac{1}{\sqrt{n}}$.

This gives the idea that we could shift the lower corner point $(-1,..., -1)$ to the origin, and the lower plane will be shifted to $x_1+...x_n=n-1$, since the distance should be $\frac{n-1}{\sqrt{n}}$. This is the plane that intersects with each axis at $n-1$.

This cut-off part can then be calculated as

$$I=\int^{n-1}_0 \int_0^{n-1-x_1} \int_0^{n-1-x_1-x_2}...\int_0^{n-1-x_1-...x_{n-1}}dx_n...dx_1$$

So the original integral is $2^n-2\cdot I=2^n-2\frac{(n-1)^n}{n!}$.