I wanted to calculate volume of the set
$$S_{d}:= \{ x_{1}, x_{2}, \ldots, x_{d} : |x_{i}| \leq 1 \text{ for each } 1 \leq i \leq d \text{ and } |x_{1} + x_{2} + \cdots + x_{d}| \leq 1\}.$$
Is it possible to calculate these for any value of $n$? I suspect one has to use calculus, but I am failing to see a solution for a while now.