GATE 2020 IN GA Q.5: solution and answer

Official answer: DSource: Official answer key, GATE 2020 IN (IIT Delhi)

1. The greatest integer (floor) function $[x]$ equals the integer $n$ for $n\le x<n+1$ and jumps up by 1 at each integer. On $[1,4]$ its graph is three horizontal steps. All of them lie above the $x$-axis ($y\ge 1>0$), so the area under $y$ is simply ${\int }_1^4[x]dx$.

2. Split the interval at the integers: $[1,2)$, $[2,3)$, $[3,4]$. On these pieces $y=1$, $y=2$ and $y=3$. At the single point $x=4$, $y=4$, but a point has zero width and adds no area.

3. Each step is a rectangle 1 unit wide, so its area equals its height: $1\times 1=1$, $1\times 2=2$ and $1\times 3=3$.

4. Add the three rectangles: $${\int }_1^4[x]dx=1+2+3=6$$

5. Quick check: the interval is 3 units wide. On $[2,4]$ we have $y\ge 2>1$, so the area is strictly more than $1\times 3=3$. Apart from the single point $x=4$, $y\le 3<4$, so the area is strictly less than $4\times 3=12$. That leaves only 4 and 6 among the options, and the exact sum is 6. The answer is option (D).