Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

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 by 123456 last updated on 14/Jun/16

what is E(x)?

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

Commented by FilupSmith 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 by enigmeyou last updated on 15/Jun/16

the floor and ceiling function

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

Commented by FilupSmith 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 by enigmeyou last updated on 15/Jun/16

good but what about k_i ?

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

Terms of Service

Privacy Policy

Contact: info@tinkutara.com