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Question Number 79903 by Pratah last updated on 29/Jan/20

Commented by jagoll last updated on 29/Jan/20

$$\mathrm{what}\mathrm{definition}\mathrm{of}\left[\mathrm{x}\right]?$$$$\mathrm{it}\mathrm{same}\mathrm{with}\lfloor \mathrm{x}\rfloor ?$$

Commented by mr W last updated on 29/Jan/20

Commented by jagoll last updated on 29/Jan/20

$$\{3.47\}=3.47-[3.47]$$$$=3.47-3=0.47$$$$\mathrm{if}\{5.75\}=5.75-[5.75]$$$$=5.75-5=0.75$$

Commented by jagoll last updated on 29/Jan/20

$$\{-7.02\}=-7-[-7.02]$$$$=-7-(-7)=0$$$$\mathrm{that}\mathrm{right}\mathrm{sir}?$$

Commented by jagoll last updated on 29/Jan/20

$$\mathrm{or}\mathrm{wrong}?$$

Commented by mr W last updated on 29/Jan/20

$$wrong!$$$$\{-7.02\}=-7.02-[-7.02]$$$$=-7.02-(-8)=0.98$$

Commented by jagoll last updated on 29/Jan/20

$$\mathrm{oo}{\mathrm{i}}^{\prime }\mathrm{m}\mathrm{wrong}\mathrm{in}[-7.02]$$$$\mathrm{so}\mathrm{if}[-4.001]=-5\mathrm{sir}$$

Commented by mr W last updated on 29/Jan/20

$$yes$$

Commented by jagoll last updated on 29/Jan/20

$$\mathrm{thank}\mathrm{you}.$$$$\mathrm{now}\lfloor \mathrm{x}\rfloor \mathrm{what}\mathrm{is}\mathrm{definition}\mathrm{sir}?$$

Commented by jagoll last updated on 29/Jan/20

$$\mathrm{floor}\mathrm{function}\mathrm{namely}?$$

Commented by mr W last updated on 29/Jan/20

$$whatyousawaboveisabout\lfloor x\rfloor .$$$$i.e.\left[x\right]=\lfloor x\rfloor .fornon-negativex,$$$$\left[x\right]always=\lfloor x\rfloor .butbecarefulwith$$$$negativex,becausetherearedifferent$$$$opinionsaboutthedefinitionof\left\{x\right\}$$$$and\left[x\right].whatyousawaboveisjust$$$$oneofthoseopinions,maybethe$$$$mostpopularone.ifyouwantto$$$$knowmore,gotogoogle!$$

Answered by mr W last updated on 30/Jan/20

$${\int }_0^2\left[x^2\right]dx$$$$={\int }_0^1\left[x^2\right]dx+{\int }_1^{\sqrt{2}}\left[x^2\right]dx+{\int }_{\sqrt{2}}^{\sqrt{3}}\left[x^2\right]dx+{\int }_{\sqrt{3}}^2\left[x^2\right]dx$$$$={\int }_0^1\left(0\right)dx+{\int }_1^{\sqrt{2}}\left(1\right)dx+{\int }_{\sqrt{2}}^{\sqrt{3}}\left(2\right)dx+{\int }_{\sqrt{3}}^2\left(3\right)dx$$$$=0+(\sqrt{2}-1)+2(\sqrt{3}-\sqrt{2})+3(2-\sqrt{3})$$$$=5-\sqrt{2}-\sqrt{3}$$$$\approx 1.854$$$$$$$$generally$$$${\int }_0^n\left[x^2\right]dx=n(n^2-1)-\underset{k=1}{\sum ^{n^2-1}}\sqrt{k}$$$$example:$$$${\int }_0^{100}\left[x^2\right]dx=100\times 9999-\underset{k=1}{\sum ^{9999}}\sqrt{k}\approx 333283.541$$

Commented by Pratah last updated on 29/Jan/20

$$\mathrm{great}$$

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