Set, Relation and FunctionHard
Question
If A denotes the area between the curve 2{y} = [x] + 1 for 0 ≤ y < 1 and the x-axis between x2 - x ≤ 0, then A is less than or equal to (where {.} and [.] are the fractional part and greatest integer functions respectively) -
Options
A.π/6
B.3/4
C.1/2
D.1/3
Solution
2{y} = [x] + 1 ....(1)
∵ 0 ≤ y < 1 ⇒ {y} = y
∴ (1) ⇒ 2y = [x] + 1
0 ≤ [x] + 1 < 2
- 1 ≤ [x] < 1
but [x] is an integer
⇒ [x] = - 1 ⇒ -1 ≤ x < 0
[x] = 0 ⇒ 0 ≤ x < 1
when x ∈ [-1, 0) ⇒ y = 0
when x ∈ [0, 1) ⇒ y =
∴ Area =
∵ 0 ≤ y < 1 ⇒ {y} = y
∴ (1) ⇒ 2y = [x] + 1
0 ≤ [x] + 1 < 2
- 1 ≤ [x] < 1
but [x] is an integer
⇒ [x] = - 1 ⇒ -1 ≤ x < 0
[x] = 0 ⇒ 0 ≤ x < 1
when x ∈ [-1, 0) ⇒ y = 0
when x ∈ [0, 1) ⇒ y =
∴ Area =
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