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For the curve 2{y} = [x] + 1 defined for 0 <= x < 1, and integrating over the region where x² - x <= 0 (i.e., 0 <= x <= 1), let A be the area between the curve and the x-axis. Then A is less than or equal to (where and [] denote fractional part and greatest integer functions respectively):

1. pi/6
2. 3/4
3. 1/2
4. 1/3

Correct answer: 1/2

Solution

For 0<=x<1: [x]=0, so 2{y}=1, meaning {y}=1/2, i.e. y = n + 1/2 for integer n. The region x²-x<=0 gives 0<=x<=1. Taking the branch closest to the x-axis (y=1/2), the area under y=1/2 from x=0 to x=1 equals 1/2. Hence A <= 1/2.

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