Introduction:
In this problem, we are required to evaluate the integral of the greatest integer function of x squared from 0 to 2.

Solution:
The greatest integer function of x squared, [x^2], takes on the value of the greatest integer less than or equal to x squared. We can rewrite the integral as follows:

∫[x^2] dx, where 0 ≤ x ≤ 2

For x ∈ [0, 1), [x^2] = 0, for x ∈ [1, √2), [x^2] = 1, and for x ∈ [√2, 2], [x^2] = 2. Therefore, we can split the integral into three parts and evaluate each part separately:

∫[x^2] dx = ∫0^1 [x^2] dx + ∫1^√2 [x^2] dx + ∫√2^2 [x^2] dx

∫0^1 [x^2] dx = ∫0^1 0 dx = 0

∫1^√2 [x^2] dx = ∫1^√2 1 dx = √2 - 1

∫√2^2 [x^2] dx = ∫√2^2 2 dx = 2(2 - √2)

Therefore, the value of the integral is:

∫[x^2] dx = 0 + √2 - 1 + 2(2 - √2) = 4 - √2

Conclusion:
In conclusion, we have evaluated the integral of the greatest integer function of x squared from 0 to 2 by splitting the integral into three parts and evaluating each part separately. The value of the integral is 4 - √2.