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Question Number 88951 by M±th+et£s last updated on 14/Apr/20

$${\int }_a^b\lceil x\rceil dx=?$$$$a,b\in R$$$$$$$$\lceil ..\rceil isceil$$

Answered by mr W last updated on 14/Apr/20

$$letA=\lceil a\rceil ,B=\lfloor b\rfloor ,C=\lceil b\rceil ,D=\lfloor a\rfloor$$$$I_1={\int }_a^b\lceil x\rceil dx$$$$={\int }_a^{\lceil a\rceil }\lceil x\rceil dx+\underset{k=\lceil a\rceil }{\sum ^{\lfloor b\rfloor -1}}{\int }_k^{k+1}\lceil x\rceil dx+{\int }_{\lfloor b\rfloor }^b\lceil x\rceil dx$$$$={\int }_a^{\lceil a\rceil }\lceil a\rceil dx+\underset{k=\lceil a\rceil }{\sum ^{\lfloor b\rfloor -1}}{\int }_k^{k+1}(k+1)dx+{\int }_{\lfloor b\rfloor }^b\lceil b\rceil dx$$$$=\lceil a\rceil (\lceil a\rceil -a)+\underset{k=\lceil a\rceil }{\sum ^{\lfloor b\rfloor -1}}(k+1)+\lceil b\rceil (b-\lfloor b\rfloor )$$$$=\lceil a\rceil (\lceil a\rceil -a)+\underset{k=\lceil a\rceil +1}{\sum ^{\lfloor b\rfloor }}k+\lceil b\rceil (b-\lfloor b\rfloor )$$$$=\lceil a\rceil (\lceil a\rceil -a)+\frac{(\lceil a\rceil +\lfloor b\rfloor +1)(\lfloor b\rfloor -\lceil a\rceil )}{2}+\lceil b\rceil (b-\lfloor b\rfloor )$$$$=A(A-a)+\frac{(A+B+1)(B-A)}{2}+C(b-B)$$$$=\frac{A^2+B^2+B-A(1+2a)+2(b-B)C}{2}$$$$I_1=\frac{\lceil a{\rceil }^2+\lfloor b{\rfloor }^2+\lfloor b\rfloor -(1+2a)\lceil a\rceil +2(b-\lfloor b\rfloor )\lceil b\rceil }{2}$$$$$$$$I_2={\int }_a^b\lfloor x\rfloor dx$$$$={\int }_a^{\lceil a\rceil }\lfloor x\rfloor dx+\underset{k=\lceil a\rceil }{\sum ^{\lfloor b\rfloor -1}}{\int }_k^{k+1}\lfloor x\rfloor dx+{\int }_{\lfloor b\rfloor }^b\lfloor x\rfloor dx$$$$={\int }_a^{\lceil a\rceil }\lfloor a\rfloor dx+\underset{k=\lceil a\rceil }{\sum ^{\lfloor b\rfloor -1}}{\int }_k^{k+1}kdx+{\int }_{\lfloor b\rfloor }^b\lfloor b\rfloor dx$$$$=\lfloor a\rfloor (\lceil a\rceil -a)+\underset{k=\lceil a\rceil }{\sum ^{\lfloor b\rfloor -1}}k+\lfloor b\rfloor (b-\lfloor b\rfloor )$$$$=\lfloor a\rfloor (\lceil a\rceil -a)+\frac{(\lceil a\rceil +\lfloor b\rfloor -1)(\lfloor b\rfloor -\lceil a\rceil )}{2}+\lfloor b\rfloor (b-\lfloor b\rfloor )$$$$=D(A-a)+\frac{(A+B-1)(B-A)}{2}+B(b-B)$$$$=\frac{2D(A-a)+A-B-A^2-B^2+2Bb}{2}$$$$I_2=\frac{2\lfloor a\rfloor (\lceil a\rceil -a)+\lceil a\rceil +(2b-1)\lfloor b\rfloor -\lceil a{\rceil }^2-\lfloor b{\rfloor }^2}{2}$$$$$$$$example:$$$$I_2={\int }_{-1.5}^{3.7}\lfloor x\rfloor dx$$$$I_2=\frac{2\lfloor -1.5\rfloor (\lceil -1.5\rceil +1.5)+\lceil -1.5\rceil +(2\times 3.7-1)\lfloor 3.7\rfloor -\lceil -1.5{\rceil }^2-\lfloor 3.7{\rfloor }^2}{2}$$$$I_2=\frac{2(-2)(-1+1.5)-1+(2\times 3.7-1)\times 3-{(-1)}^2-3^2}{2}$$$$I_2=\frac{6.2}{2}=3.1$$$$$$$$I_2={\int }_{-2}^4\lfloor x\rfloor dx$$$$I_2=\frac{2\lfloor -2\rfloor (\lceil -2\rceil +2)+\lceil -2\rceil +(2\times 4-1)\lfloor 4\rfloor -\lceil -2{\rceil }^2-\lfloor 4{\rfloor }^2}{2}$$$$I_2=\frac{-2+28-4-16}{2}=3$$

Commented by M±th+et£s last updated on 14/Apr/20

$$iamverrygreatfulltoyouandpretiatewhat$$$$haveyoudoneforme$$

Commented by M±th+et£s last updated on 14/Apr/20

$$sirdoyouhavepdfordoyouhaveanyexplains$$$$forthissubjectbecauseiamastudent$$$$andiwanttolearn$$

Commented by M±th+et£s last updated on 14/Apr/20

$$imeanfloorfunction$$

Commented by mr W last updated on 14/Apr/20

$$i{don}^{\prime }thaveanybook.but{it}^{\prime }sjusta$$$$functiondefinition,thereisnot$$$$muchtoread.$$

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