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Question Number 88951 by M±th+et£s last updated on 14/Apr/20
∫_a ^b ⌈x⌉ dx=? a,b∈R ⌈..⌉ is ceil
∫ab⌈x⌉dx=?a,b∈R⌈..⌉isceil
Answered by mr W last updated on 14/Apr/20
let A=⌈a⌉, B=⌊b⌋, C=⌈b⌉, D=⌊a⌋ I_1 =∫_a ^b ⌈x⌉ dx =∫_a ^(⌈a⌉) ⌈x⌉ dx+Σ_(k=⌈a⌉) ^(⌊b⌋−1) ∫_k ^(k+1) ⌈x⌉ dx+∫_(⌊b⌋) ^b ⌈x⌉ dx =∫_a ^(⌈a⌉) ⌈a⌉ dx+Σ_(k=⌈a⌉) ^(⌊b⌋−1) ∫_k ^(k+1) (k+1) dx+∫_(⌊b⌋) ^b ⌈b⌉ dx =⌈a⌉(⌈a⌉−a)+Σ_(k=⌈a⌉) ^(⌊b⌋−1) (k+1)+⌈b⌉(b−⌊b⌋) =⌈a⌉(⌈a⌉−a)+Σ_(k=⌈a⌉+1) ^(⌊b⌋) k+⌈b⌉(b−⌊b⌋) =⌈a⌉(⌈a⌉−a)+(((⌈a⌉+⌊b⌋+1)(⌊b⌋−⌈a⌉))/2)+⌈b⌉(b−⌊b⌋) =A(A−a)+(((A+B+1)(B−A))/2)+C(b−B) =((A^2 +B^2 +B−A(1+2a)+2(b−B)C)/2) I_1 =((⌈a⌉^2 +⌊b⌋^2 +⌊b⌋−(1+2a)⌈a⌉+2(b−⌊b⌋)⌈b⌉)/2) I_2 =∫_a ^b ⌊x⌋ dx =∫_a ^(⌈a⌉) ⌊x⌋ dx+Σ_(k=⌈a⌉) ^(⌊b⌋−1) ∫_k ^(k+1) ⌊x⌋ dx+∫_(⌊b⌋) ^b ⌊x⌋ dx =∫_a ^(⌈a⌉) ⌊a⌋ dx+Σ_(k=⌈a⌉) ^(⌊b⌋−1) ∫_k ^(k+1) k dx+∫_(⌊b⌋) ^b ⌊b⌋ dx =⌊a⌋(⌈a⌉−a)+Σ_(k=⌈a⌉) ^(⌊b⌋−1) k+⌊b⌋(b−⌊b⌋) =⌊a⌋(⌈a⌉−a)+(((⌈a⌉+⌊b⌋−1)(⌊b⌋−⌈a⌉))/2)+⌊b⌋(b−⌊b⌋) =D(A−a)+(((A+B−1)(B−A))/2)+B(b−B) =((2D(A−a)+A−B−A^2 −B^2 +2Bb)/2) I_2 =((2⌊a⌋(⌈a⌉−a)+⌈a⌉+(2b−1)⌊b⌋−⌈a⌉^2 −⌊b⌋^2 )/2) example: I_2 =∫_(−1.5) ^(3.7) ⌊x⌋ dx I_2 =((2⌊−1.5⌋(⌈−1.5⌉+1.5)+⌈−1.5⌉+(2×3.7−1)⌊3.7⌋−⌈−1.5⌉^2 −⌊3.7⌋^2 )/2) I_2 =((2(−2)(−1+1.5)−1+(2×3.7−1)×3−(−1)^2 −3^2 )/2) I_2 =((6.2)/2)=3.1 I_2 =∫_(−2) ^4 ⌊x⌋ dx I_2 =((2⌊−2⌋(⌈−2⌉+2)+⌈−2⌉+(2×4−1)⌊4⌋−⌈−2⌉^2 −⌊4⌋^2 )/2) I_2 =((−2+28−4−16)/2)=3
letA=⌈a⌉,B=⌊b⌋,C=⌈b⌉,D=⌊a⌋I1=∫ab⌈x⌉dx=∫a⌈a⌉⌈x⌉dx+∑⌊b⌋−1k=⌈a⌉∫kk+1⌈x⌉dx+∫⌊b⌋b⌈x⌉dx=∫a⌈a⌉⌈a⌉dx+∑⌊b⌋−1k=⌈a⌉∫kk+1(k+1)dx+∫⌊b⌋b⌈b⌉dx=⌈a⌉(⌈a⌉−a)+∑⌊b⌋−1k=⌈a⌉(k+1)+⌈b⌉(b−⌊b⌋)=⌈a⌉(⌈a⌉−a)+∑⌊b⌋k=⌈a⌉+1k+⌈b⌉(b−⌊b⌋)=⌈a⌉(⌈a⌉−a)+(⌈a⌉+⌊b⌋+1)(⌊b⌋−⌈a⌉)2+⌈b⌉(b−⌊b⌋)=A(A−a)+(A+B+1)(B−A)2+C(b−B)=A2+B2+B−A(1+2a)+2(b−B)C2I1=⌈a⌉2+⌊b⌋2+⌊b⌋−(1+2a)⌈a⌉+2(b−⌊b⌋)⌈b⌉2I2=∫ab⌊x⌋dx=∫a⌈a⌉⌊x⌋dx+∑⌊b⌋−1k=⌈a⌉∫kk+1⌊x⌋dx+∫⌊b⌋b⌊x⌋dx=∫a⌈a⌉⌊a⌋dx+∑⌊b⌋−1k=⌈a⌉∫kk+1kdx+∫⌊b⌋b⌊b⌋dx=⌊a⌋(⌈a⌉−a)+∑⌊b⌋−1k=⌈a⌉k+⌊b⌋(b−⌊b⌋)=⌊a⌋(⌈a⌉−a)+(⌈a⌉+⌊b⌋−1)(⌊b⌋−⌈a⌉)2+⌊b⌋(b−⌊b⌋)=D(A−a)+(A+B−1)(B−A)2+B(b−B)=2D(A−a)+A−B−A2−B2+2Bb2I2=2⌊a⌋(⌈a⌉−a)+⌈a⌉+(2b−1)⌊b⌋−⌈a⌉2−⌊b⌋22example:I2=∫−1.53.7⌊x⌋dxI2=2⌊−1.5⌋(⌈−1.5⌉+1.5)+⌈−1.5⌉+(2×3.7−1)⌊3.7⌋−⌈−1.5⌉2−⌊3.7⌋22I2=2(−2)(−1+1.5)−1+(2×3.7−1)×3−(−1)2−322I2=6.22=3.1I2=∫−24⌊x⌋dxI2=2⌊−2⌋(⌈−2⌉+2)+⌈−2⌉+(2×4−1)⌊4⌋−⌈−2⌉2−⌊4⌋22I2=−2+28−4−162=3
Commented by M±th+et£s last updated on 14/Apr/20
i am verry greatfull to you and pretiate what have you done for me
iamverrygreatfulltoyouandpretiatewhathaveyoudoneforme
Commented by M±th+et£s last updated on 14/Apr/20
sir do you have pdf or do you have any explains for this subject because i am a student and i want to learn
sirdoyouhavepdfordoyouhaveanyexplainsforthissubjectbecauseiamastudentandiwanttolearn
Commented by M±th+et£s last updated on 14/Apr/20
i mean floor function
imeanfloorfunction
Commented by mr W last updated on 14/Apr/20
i don′t have any book. but it′s just a function definition, there is not much to read.
idon′thaveanybook.butit′sjustafunctiondefinition,thereisnotmuchtoread.

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