Math Question and Answers


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 58648 by rahul 19 last updated on 27/Apr/19

Commented by rahul 19 last updated on 27/Apr/19

$A n s : b, d .$ $i^{'} m g e t t i n g t h a t i t c a n n o t e x c e e d 2 .$
Commented by maxmathsup by imad last updated on 27/Apr/19

$w e h a v e \int_{a}^{b} ∣ f (x) ∣∣ g (x) ∣ d x ⩽ {(\int_{a}^{b} f^{2} (x) d x)}^{\frac{1}{2}} . (\int_{a}^{b} g^{2} (x) d x) (c a u c h y s h w a r z)$ $l e t t a k e f (x) = \sqrt{1 + x} a n d g (x) = \sqrt{1 + x^{3}} d x \Rightarrow$ $\int_{0}^{1} \sqrt{1 + x} \sqrt{1 + x^{3}} d x ⩽ \sqrt{\int_{0}^{1} (1 + x) d x} . \sqrt{\int_{0}^{1} (1 + x^{3}) d x}$ $b u t \int_{0}^{1} (1 + x) d x = {[x + \frac{x^{2}}{2}]}_{0}^{1} = \frac{3}{2}$ $\int_{0}^{1} (1 + x^{3}) d x = {[x + \frac{x^{4}}{4}]}_{0}^{1} = \frac{5}{4} \Rightarrow I ⩽ \sqrt{\frac{3}{2}} . \sqrt{\frac{5}{4}} \Rightarrow I ⩽ \sqrt{\frac{15}{8}}$ $s o t h e a n s w e r i s (d) .$
Commented by rahul 19 last updated on 28/Apr/19

$t h a n k U p r o f . A b d o$
Commented by maxmathsup by imad last updated on 30/Apr/19

$y o u a r e w e l c o m e .$
	Answered by tanmay last updated on 27/Apr/19
	$((1+x)/2)≥(1×x)^(1/2) ((1+x^3 )/2)≥(1×x^3 )^(1/2) (1+x)(1+x^3 )≥4(x.x^3 )^(1/2) (1+x)(1+x^3 )≥4x^2 (√((1+x)(1+x^3 ))) ≥2x ((1+x+1+x^3 )/2)≥{(1+x)(1+x^3 )}^(1/2) ((2+x+x^3 )/2)≥(√((1+x)(1+x^3 ))) so ∫_0 ^1 ((2+x+x^3 )/2)dx≥∫_0 ^1 (√((1+x)(1+x^3 ))) dx≥∫_0 ^1 2xdx (1/2)∣2x+(x^2 /2)+(x^4 /4)∣_0 ^1 ≥I≥2×∣(x^2 /2)∣_0 ^1 (1/2)×((11)/4)≥I≥1 ((11)/8)≥I≥1 ((11)/8)=1.375 and (√((15)/8)) =(√(1.875)) =1.369≈1.37$
	$\frac{1 + x}{2} ⩾ {(1 \times x)}^{\frac{1}{2}}$ $\frac{1 + x^{3}}{2} ⩾ {(1 \times x^{3})}^{\frac{1}{2}}$ $(1 + x) (1 + x^{3}) ⩾ 4 {(x . x^{3})}^{\frac{1}{2}}$ $(1 + x) (1 + x^{3}) ⩾ 4 x^{2}$ $\sqrt{(1 + x) (1 + x^{3})} ⩾ 2 x$ $\frac{1 + x + 1 + x^{3}}{2} ⩾ {(1 + x) (1 + x^{3})}^{\frac{1}{2}}$ $\frac{2 + x + x^{3}}{2} ⩾ \sqrt{(1 + x) (1 + x^{3})}$ $s o$ $\int_{0}^{1} \frac{2 + x + x^{3}}{2} d x ⩾ \int_{0}^{1} \sqrt{(1 + x) (1 + x^{3})} d x ⩾ \int_{0}^{1} 2 x d x$ $\frac{1}{2} ∣ 2 x + \frac{x^{2}}{2} + \frac{x^{4}}{4} ∣_{0}^{1} ⩾ I ⩾ 2 \times ∣ \frac{x^{2}}{2} ∣_{0}^{1}$ $\frac{1}{2} \times \frac{11}{4} ⩾ I ⩾ 1$ $\frac{11}{8} ⩾ I ⩾ 1$ $\frac{11}{8} = 1.375 a n d \sqrt{\frac{15}{8}} = \sqrt{1.875} = 1.369 \approx 1.37$
	Commented by tanmay last updated on 28/Apr/19

	Commented by rahul 19 last updated on 28/Apr/19

	$S i r, i f I d o t h e s a m e p r o c e d u r e a s i n Q . 58626$ $t h e n i w i l l g e t :$ $f (0) = 1 \Rightarrow m = 1$ $f (1) = 2 \Rightarrow M = 2$ $\Rightarrow\Rightarrow 1 < I < 2 .$ ${W h a t}^{'} s w r o n g i n t h i s m e t h o d ?$
	Commented by tanmay last updated on 28/Apr/19

	$i a m c o r e l a t i n g w i t h m e a n v a l u e t h e o r e m o f$ $i n t r e g a l$
	Commented by tanmay last updated on 28/Apr/19

	Commented by tanmay last updated on 28/Apr/19

	Commented by rahul 19 last updated on 29/Apr/19

	$s t i l l c o n f u s e d . . .$ $w h y c a n w e a p p l y i n Q . 58626 t h e n ?$ $(i n i n t e r v a l 1 t o 3 g i v e n f u n c t i o n d o e s$ $n o t h a v e m i n / m a x . v a l u e .)$
	Answered by tanmay last updated on 27/Apr/19

	Commented by tanmay last updated on 27/Apr/19

	$i n b o o k s a n s w e r i s a b o v e a l o n g w i t h f o r m u l a$ $h o w e v e r \frac{11}{8} = 1.375 \leftarrow w h a t i c a l c u l a t e d$ $\sqrt{\frac{15}{8}} = \sqrt{1.875} = 1.369 \leftarrow w h a t g i v e n i n b o o k$
	Commented by rahul 19 last updated on 28/Apr/19

	$t h a n k u s i r .$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum