Answer
$$1$$
Work Step by Step
Our aim is to integrate the triple integral as follows:
$$ Average=\int^1_0 \int^1_0 \int^1_0 (x^2+y^2+z^2) \space dz \space dy \space dx \\=\int^1_0 \int^1_0 (x^2+y^2+\dfrac{1}{3}) \space dy \space dx \\=\int^1_0 (x^2+\dfrac{2}{3})\space dx \\=1$$
