Let $f(x_1,x_2,\cdots ,x_n)=1, \qquad 0<x_1,x_2,\cdots,x_n<1$. I would like to evaluate the integral
$$\underset{\left\{\begin{array}{cc} 0<x_1<x_2<\cdots<x_n<1,\\ x_1>a,x_2>x_1+a,\cdots,x_n>x_{n-1}+a \end{array}\right\}}{\int \int \cdots \int} f(x_1,x_2,\cdots ,x_n)\,dx_n\, dx_{n-1}\cdots dx_1$$ for fixed $a$.
I wonder if you could check my answer and help me to correct it with explanation in the case that it is false. $$\int_{a}^{1} \int_{x_{1}+a}^{1} \cdots \int_{x_{n-2}+a}^{1} \int_{x_{n-1}+a}^1 f(x_1,x_2,\cdots ,x_n)\,dx_n\, dx_{n-1}\cdots dx_1$$