Question

Let
S = {(x, y, z) ∈ \(\R^3\) : x² + y² + z² < 1}.
Then, the value of
\(\frac{1}{\pi}\iiint_s\left((x-2y+z)^2+(2x-y-z)+(x-y+2z)^2\right)dxdydz\)
equals ________ (rounded off to two decimal places).

Answer 1

Correct Answer: 4.7

The problem involves evaluating a triple integral over the unit sphere \( S \). By simplifying the integrand and using symmetry properties of the sphere, the result of the integral is approximately 4.70 when rounded to two decimal places. Thus, the correct answer is 4.70.