Question

Area enclosed by [x]² = [y]² for 1 ≤ x ≤ 4

• A. 8 sq. units
• B. 6 sq. units
• C. 4 sq. units
• D. 12 sq. units

Answer 1

Answer: (B)

6 sq. units

Answer 2

The equation [x]² = [y]² implies [y] = ±[x]. We consider the intervals for x:

1. For 1 ≤ x < 2: [x] = 1. Then [y] = ±1, which means y ∈ [-1, 0) ∪ [1, 2). The area in this strip is 1 (width of x) × (1+1) (height of y) = 2.
2. For 2 ≤ x < 3: [x] = 2. Then [y] = ±2, which means y ∈ [-2, -1) ∪ [2, 3). The area in this strip is 1 (width of x) × (1+1) (height of y) = 2.
3. For 3 ≤ x < 4: [x] = 3. Then [y] = ±3, which means y ∈ [-3, -2) ∪ [3, 4). The area in this strip is 1 (width of x) × (1+1) (height of y) = 2. The total area is the sum of the areas from these intervals: 2 + 2 + 2 = 6 sq. units.