Integral Part Integer

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What Is The Value of

< m a t h > ∫ 2 5 ∫ 1 3 ⌊ x + y ⌋ d x d y < m a t h > {\LARGE{<math> \int_2^5 \int_1^3 \left \lfloor{x+y}\right \rfloor dx dy<math>}}

Note :


The answer is 30.

Gabriel Merces by Gabriel Merces , 23, Brazil

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1 solution

Cody Johnson
Mar 14, 2014

By casing it on the fractional part of y y , we get

∫ 2 5 ∫ 1 3 ⌊ x + y ⌋ d x d y = ∑ n = 3 5 ∫ 0 1 ( 2 n + 1 + 2 x ) d x = ∑ n = 3 5 ( 2 n + 2 ) = 30 \int_2^5\int_1^3\left\lfloor x+y\right\rfloor\,dx\,dy=\sum_{n=3}^5\int_0^1(2n+1+2x)\,dx=\sum_{n=3}^5(2n+2)=\boxed{30}

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