Evaluate $\int ...\int_{R_n}dV_n(x_1^2 + x_2^2 + ... + x_n^2)$ , where $n$ and $R_n$ is defined in the body of this question.

The integral is

$$\int_{R_n} \sum_{i=1}^{n} x_i^2  \, \mathrm{d}V = \sum_{i=1}^{n} \int_{R_n} x_i^2 \, \mathrm{d}V = \sum_{i=1}^{n} \int_{0}^{1} x_i^2 \, \mathrm{d}x_i = \frac{n}{3}.$$

Here, we used the linearity of the integral to split the integral into a sum, and Fubini's theorem to write the integral $\int_{R_n} x_i^2 \, \mathrm{d}V = (\int_{0}^{1} \, \mathrm{d}x_1)(\int_{0}^{1} \, \mathrm{d}x_2)\cdots(\int_{0}^1 x_i^2 \, \mathrm{d}x_i)\cdots(\int_{0}^{1} \, \mathrm{d}x_n) = \int_{0}^1 x_i^2 \, \mathrm{d}V$