GATE 2026 MA – Question 64
On D=[0,2]×[0,2], let f(t)=⌈t⌉, the smallest integer at least t. Find $\iint_D f(x+y)dx\,dy$ (integer).
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Correct answer: 10 to 10
Explanation
Away from the measure-zero integer-sum lines, ⌈s⌉+⌈4−s⌉=5. The area-preserving map (x,y)→(2−x,2−y) changes x+y to 4−(x+y) and leaves the square invariant. Adding the original and transformed integrals gives 2I=5×area(D)=20. Thus **I=10**.