Question

If [x] denotes the greatest integer function, then find `int_0^(3/2) [x^2] dx`

Answer 1

`int_0^(3/2) [x^2] dx`

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

For 1 ≤ x < `sqrt(2)`, 1 ≤ x² < 2, hence [x²] = 1

For `sqrt(2) ≤ x < 3/2, 2 ≤ x^2 < 9/4`, hence [x²] = 2

`int_0^(3/2) [x^2] dx = int_0^1 0dx + int_1^sqrt(2) 1dx + int_sqrt(2)^(3/2) 2dx`

= `0[x]_0^1 + 1[x]_1^sqrt(2) + 2[x]_sqrt(2)^(3/2)`

= `0 + sqrt(2) - 1 + 3 - 2sqrt(2)`

= `2 - sqrt(2)`