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Question Number 6113 by enigmeyou last updated on 14/Jun/16

∫_0 ^(100) E(x)dx=?

$$\int_{\mathrm{0}} ^{\mathrm{100}} {E}\left({x}\right){dx}=? \\ $$ $$ \\ $$

Commented by123456 last updated on 14/Jun/16

what is E(x)?

$$\mathrm{what}\:\mathrm{is}\:\mathrm{E}\left({x}\right)? \\ $$

Commented byFilupSmith last updated on 15/Jun/16

Floor func. ∫_a ^( b) ⌊x⌋dx, a,b∈Z = rectangle =base×height height a≤x<a+1 ⇒ ⌊x⌋= a a+1≤x<a+2 ⇒ ⌊x⌋= a+1 ... b−1≤x<b ⇒ ⌊x⌋ = b−1 base=1 ∫_a ^( b) ⌊x⌋dx=a+(a+1)+...+(b−2)+(b−1) first term=a last term = b−1 ∴ ∫_a ^( b) ⌊x⌋dx = (((b−1)−a)/2)(a+b−1) ???

$$\mathrm{Floor}\:\mathrm{func}. \\ $$ $$\int_{{a}} ^{\:{b}} \lfloor{x}\rfloor{dx},\:\:\:{a},{b}\in\mathbb{Z}\:=\:{rectangle}\:={base}×{height} \\ $$ $$\boldsymbol{{height}} \\ $$ $${a}\leqslant{x}<{a}+\mathrm{1}\:\:\Rightarrow\:\:\lfloor{x}\rfloor=\:{a} \\ $$ $${a}+\mathrm{1}\leqslant{x}<{a}+\mathrm{2}\:\:\Rightarrow\:\:\lfloor{x}\rfloor=\:{a}+\mathrm{1} \\ $$ $$... \\ $$ $${b}−\mathrm{1}\leqslant{x}<{b}\:\:\Rightarrow\:\:\lfloor{x}\rfloor\:=\:{b}−\mathrm{1} \\ $$ $$\boldsymbol{{base}}=\mathrm{1} \\ $$ $$\int_{{a}} ^{\:{b}} \lfloor{x}\rfloor{dx}={a}+\left({a}+\mathrm{1}\right)+...+\left({b}−\mathrm{2}\right)+\left({b}−\mathrm{1}\right) \\ $$ $${first}\:{term}={a} \\ $$ $${last}\:{term}\:=\:{b}−\mathrm{1} \\ $$ $$\therefore\:\int_{{a}} ^{\:{b}} \lfloor{x}\rfloor{dx}\:=\:\frac{\left({b}−\mathrm{1}\right)−{a}}{\mathrm{2}}\left({a}+{b}−\mathrm{1}\right) \\ $$ $$??? \\ $$

Commented byenigmeyou last updated on 15/Jun/16

the floor and ceiling function

$${the}\:{floor}\:{and}\:{ceiling}\:{function} \\ $$

Commented byFilupSmith last updated on 15/Jun/16

Ceiling func. ∫_a ^( b) ⌈x⌉dx, a,b∈Z a≤x<a+1 ⇒ ⌈x⌉=a+1 ... b−1≤x<b ⇒ ⌈x⌉= b ∫_a ^( b) ⌈x⌉dx=(a+1)+(a+2)+...+(b−1)+b first term=a+1 last term=b ∴∫_a ^( b) ⌈x⌉dx = ((b−(a+1))/2)(a+b+1) ???

$$\mathrm{Ceiling}\:\mathrm{func}. \\ $$ $$\int_{{a}} ^{\:{b}} \lceil{x}\rceil{dx},\:\:\:{a},{b}\in\mathbb{Z} \\ $$ $${a}\leqslant{x}<{a}+\mathrm{1}\:\:\Rightarrow\:\:\lceil{x}\rceil={a}+\mathrm{1} \\ $$ $$... \\ $$ $${b}−\mathrm{1}\leqslant{x}<{b}\:\Rightarrow\:\lceil{x}\rceil=\:{b} \\ $$ $$\int_{{a}} ^{\:{b}} \lceil{x}\rceil{dx}=\left({a}+\mathrm{1}\right)+\left({a}+\mathrm{2}\right)+...+\left({b}−\mathrm{1}\right)+{b} \\ $$ $${first}\:{term}={a}+\mathrm{1} \\ $$ $${last}\:{term}={b} \\ $$ $$ \\ $$ $$\therefore\int_{{a}} ^{\:{b}} \lceil{x}\rceil{dx}\:=\:\frac{{b}−\left({a}+\mathrm{1}\right)}{\mathrm{2}}\left({a}+{b}+\mathrm{1}\right) \\ $$ $$??? \\ $$

Answered by petmill last updated on 15/Jun/16

∫_0 ^(100) E(x)dx=Σ_(i=o) ^(99) (x_(i+1) −x_i )k_i =1+2+....+99=4950

$$\int_{\mathrm{0}} ^{\mathrm{100}} {E}\left({x}\right){dx}=\underset{{i}={o}} {\overset{\mathrm{99}} {\sum}}\left({x}_{{i}+\mathrm{1}} −{x}_{{i}} \right){k}_{{i}} =\mathrm{1}+\mathrm{2}+....+\mathrm{99}=\mathrm{4950} \\ $$

Commented byenigmeyou last updated on 15/Jun/16

good but what about k_i ?

$${good}\:{but}\:{what}\:{about}\:{k}_{{i}} ? \\ $$

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