What is the area of the region defined by [x]² = [y]², where x belongs to [1, 3] and [.] denotes the greatest integer function? The area is less than which of the following values?

Question

Accepted Answer

5. The condition [x]² = [y]² implies [y] = +[x] or [y] = -[x]. For x in [1,2): [x]=1, so [y]=1 (y in [1,2)) or [y]=-1 (y in [-1,0)). Each strip has width 1 (in x) and height 1 (in y), area = 1 each. For x in [2,3): [x]=2, so [y]=2 (y in [2,3)) or [y]=-2 (y in [-2,-1)), again area = 1 each. At x=3 it is a single line of measure zero. Total area = 4. Since 4 < 5 and 4 < 10, both options A (5) and C (10) are correct statements. However option A (less than 5) is the tightest and most meaningful correct answer.

What is the area of the region defined by [x]² = [y]², where x belongs to [1, 3] and [.] denotes the greatest integer function? The area is less than which of the following values?

Solution

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