prove that (1/e) ≤ ∫_0 ^1 e^(−(x−[x])^2 ) dx≤1.


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Question Number 30442 by abdo imad last updated on 22/Feb/18

$p r o v e t h a t \frac{1}{e} ⩽ \int_{0}^{1} e^{- {(x - [x])}^{2}} d x ⩽ 1 .$
Commented by alex041103 last updated on 22/Feb/18

$Y o u c a n s e e t h a t i f x \in [0, 1), t h e n$ $[x] = 0 \Rightarrow x - [x] = x$ $\Rightarrow \int_{0}^{1} e^{- {(x - [x])}^{2}} d x = \int_{0}^{1} e^{- x^{2}} d x$
Commented by abdo imad last updated on 24/Feb/18

$w e h a v e [x] ⩽ x < [x] + 1 \Rightarrow 0 ⩽ x - [x] < 1 \Rightarrow$ $0 ⩽ {(x - [x])}^{2} < 1 \Rightarrow - 1 < - {(x - [x])}^{2} ⩽ 0 \Rightarrow$ $e^{- 1} < e^{- {(x - [x])}^{2}} ⩽ 1 \Rightarrow \frac{1}{e} ⩽ \int_{0}^{1} e^{- {(x - [x])}^{2}} d x ⩽ 1 .$
	Answered by alex041103 last updated on 22/Feb/18

	Commented by alex041103 last updated on 22/Feb/18

	$T h e o r a n g e f u n c t i o n i s e^{- x^{2}}$ $T h e b l u e a r e a i s t h e i n t e g r a l .$ $B e c a u s e o f t h a t i n t e g r a l s r e p r e s e n t$ $t h e a r e a b e t w e e n \vec{x} a n d t h e f u c t i o n,$ $w e s e e t h a t$ $\int_{0}^{1} e^{- x^{2}} d x ⩽ r e d a r e a = 1^{2} = 1$ $\Rightarrow \int_{0}^{1} e^{- x^{2}} d x ⩽ 1$
	Commented by rahul 19 last updated on 23/Feb/18

	$r u writing ans . in flight, as aeroplane$ $mode is on ?$
	Commented by alex041103 last updated on 24/Feb/18

	$N o . I w a s a b r o a d, s o t h e r e {i s n}^{'} t a n y$ $p o i n t i n u s i n g m y b a t t e r y f o r m o b i l e$ $s e r v i c e .$
	Commented by rahul 19 last updated on 24/Feb/18

	$hmmm, fine .$
	Answered by alex041103 last updated on 22/Feb/18

	Commented by alex041103 last updated on 22/Feb/18

	$B y f o l l o w i n g t h e s a m e l o g i c w e c a n s e e$ $t h a t :$ $\int_{0}^{1} e^{- x^{2}} d x ⩾ g r e e n - i s h a r e a = 1 \times e^{- {(1)}^{2}} = \frac{1}{e}$ $\Rightarrow \int_{0}^{1} e^{- x^{2}} d x ⩾ \frac{1}{e}$
	Commented by abdo imad last updated on 23/Feb/18

	$y o u r a n s w e r i s c o r r e c t s i r b y u s i n g g r a p h t h a n k s s i r . . .$
	Answered by alex041103 last updated on 22/Feb/18

	$O r a s a c o n c l u s i o n :$ $\frac{1}{e} ⩽ \int_{0}^{1} e^{- {(x - [x])}^{2}} d x ⩽ 1 .$ $Q u e s t i o n s ?$
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