Question

Integrate [x^2] (Greatest Integer Function) with limit 0 to 2.

Answer 1

5 answers · Mathematics

Best Answer

f(x) = [x²]. . . . (floor function of x²)

For 0 ≤ x < 1, 0 ≤ x² < 1, hence [x²] = 0

For 1 ≤ x < √2, 1 ≤ x² < 2, hence {x²] = 1

For √2 ≤ x < √3, 2 ≤ x² < 3, hence [x²] = 2

For √3 ≤ x < 2, 3 ≤ x² < 2, hence [x²] = 3

2. . . . . . .1. . . . . .√2. . . . . √3. . . . . .2

∫ [x²]dx = ∫ [x²]dx + ∫ [x²]dx + ∫ [x²]dx + ∫ [x²]dx

0. . . . . . 0. . . . . . 1. . . . . . √2. . . . .√3

2. . . . . . .1. . . . √2. . . . .√3. . . . . 2

∫ [x²]dx = ∫ 0 dx + ∫ 1 dx + ∫ 2 dx + ∫ 3 dx

0. . . . . . 0. . . . . 1. . . . . √2. . . . .√3

2

∫ [x²]dx = 0*(2 - 0) + 1*(√2 - 1) + 2*(√3 - √2) + 3*(2 - √3)

0

2

∫ [x²]dx = 0 + √2 - 1 + 2√3 - 2√2 + 6 - 3√3

0

2

∫ [x²]dx = 5 - √2 - √3

0