A calculus problem by Hobart Pao

Calculus Level 4

I = ∫ 0 ∞ ∫ 0 ∞ sin ⁡ x sin ⁡ y x y d y d x \large \mathscr{I} = \displaystyle \int \limits_{0}^{\infty} \int \limits_{0}^{\infty} \dfrac{\sin x \sin y}{xy} \, dy \, dx

Find ⌊ 1000 I ⌋ \left \lfloor 1000\mathscr{I} \right \rfloor


The answer is 2467.

Hobart Pao by Hobart Pao , 23, USA

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1 solution

Hobart Pao
May 3, 2017

I = ∫ 0 ∞ ∫ 0 ∞ sin ⁡ x sin ⁡ y x y d y d x = ∫ 0 ∞ sin ⁡ x x [ ∫ 0 ∞ sin ⁡ y y d y ] d x = ( ∫ 0 ∞ sin ⁡ x x d x ) ( ∫ 0 ∞ sin ⁡ y y d y ) = ( π 2 ) 2 = π 2 4 ≈ 2 , 467 \mathscr{I} = \displaystyle \int \limits_0^{\infty}\int \limits_0^{\infty}\dfrac{\sin x \sin y }{xy} \, dy \, dx = \int \limits_0^{\infty} \dfrac{\sin x}{x} \left[ \int \limits_0^{\infty} \dfrac{\sin y}{y} \, dy \right] \, dx = \left( \int \limits_0^{\infty} \dfrac{\sin x}{x} \, dx \right) \left( \int \limits_0^{\infty} \dfrac{\sin y}{y} \, dy \right) = \left( \dfrac{\pi}{2} \right)^2 = \dfrac{\pi^2}{4} \approx 2,467

⌊ 1000 ⋅ 2 , 467... ⌋ = 2467 \left \lfloor 1000 \cdot 2,467... \right \rfloor = \boxed{2467}

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice solution (+1). Did the same way

Rahil Sehgal - 4 years, 1 month ago

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