Evaluation of Definite Integrals

For x less than 1, abs(x - sqrtx) = sqrtx -x.
For x greater than 1, abs(x-sqrtx) = x-sqrtx.
Therefore, the integral must be split into 2 parts, from 0 to 1 and from 1 to 2.
S(sqrtx - x)dx = [2x^(3/2)/3 - x^2/2 ] from 1 to 0 = 2/3 - 1/2 = 1/6.
S(x- sqrtx)dx = [x^2/2 - 2x^(3/2)/3] from 2 to 1= (2 - 2sqrt8/3 -1/2 + 2/3)
= 2-4sqrt2/3 + 1/6
Adding them together, we get 7/3 - 4sqrt2/3 = (7-4sqrt2)/3.

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哈，又慢左步添~