Find integration of $$\int_{-2}^{2}\lfloor x^2-1\rfloor~dx$$ where $\lfloor . \rfloor$ is Box function i.e. greatest integer function . More explicitely [x]= greatest integer not greater than x.
Find integration of box function
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real-analysis
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1Your "Box function" is formally called the "[floor function](http://en.wikipedia.org/wiki/Floor_and_ceiling_functions)". – 2012-12-19
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0The answer to [this question](http://math.stackexchange.com/questions/261482/if-for-x-in-r-phi-x-denotes-the-integer-closest-to-x-then-int-10) should point you in the right direction. – 2012-12-19
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0Oops! Is that the floor function? Then I shall erase my answer... – 2012-12-19
1 Answers
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$$ \begin{align} \int_{-2}^{2}\lfloor x^2-1\rfloor\,\mathrm{d}x &=2\int_0^{2}\lfloor x^2-1\rfloor\,\mathrm{d}x\\ &=2\int_0^1-1\,\mathrm{d}x +2\int_1^{\sqrt{2}}0\,\mathrm{d}x +2\int_{\sqrt2}^{\sqrt3}1\,\mathrm{d}x +2\int_{\sqrt3}^22\,\mathrm{d}x\\[6pt] &=-2+2(\sqrt3-\sqrt2)+4(2-\sqrt3)\\[12pt] &=6-2\sqrt2-2\sqrt3 \end{align} $$
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0Thanks for your help. Now I got it. – 2012-12-19