Is $\displaystyle f(x,y) = (\sin(x) \sin(y)) / ({x^2 + y^2})$ integrable over $\displaystyle (-\pi/2,\pi/2) \times (-\pi/2,\pi/2)$?

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- Mar 7th 2009, 09:24 AMAmanda1990[SOLVED] Integrability
Is $\displaystyle f(x,y) = (\sin(x) \sin(y)) / ({x^2 + y^2})$ integrable over $\displaystyle (-\pi/2,\pi/2) \times (-\pi/2,\pi/2)$?

- Mar 7th 2009, 09:41 AMLaurent
Here's my answer: since $\displaystyle |\sin x\sin y|\leq |x||y|\leq \frac{1}{2}(x^2+y^2)$, we have $\displaystyle |f(x,y)|\leq \frac{1}{2}$. Thus $\displaystyle f$ is bounded and continuous, except perhaps at the point 0 (doesn't matter), so that it is integrable on any bounded subset of $\displaystyle \mathbb{R}^2$.