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Question Number 138611 by liberty last updated on 15/Apr/21

Given a function f where   f(x)≥ 0for ∀x∈R. If the area  U = { (x,y)∣0≤2y≤f(x), −6≤x≤−2}  is u and the area V={(x,y)∣0≤y≤f(x),−2≤x≤0}  is v then what the value of  ∫_1 ^2  4x f(2x^2 −8) dx .  (A) 5u+4v     (D)2u+v  (B) 4u+3v    (E) u+v  (C) 3u+2v

$${Given}\:{a}\:{function}\:{f}\:{where}\: \\ $$$${f}\left({x}\right)\geqslant\:\mathrm{0}{for}\:\forall{x}\in\mathbb{R}.\:{If}\:{the}\:{area} \\ $$$${U}\:=\:\left\{\:\left({x},{y}\right)\mid\mathrm{0}\leqslant\mathrm{2}{y}\leqslant{f}\left({x}\right),\:−\mathrm{6}\leqslant{x}\leqslant−\mathrm{2}\right\} \\ $$$${is}\:{u}\:{and}\:{the}\:{area}\:{V}=\left\{\left({x},{y}\right)\mid\mathrm{0}\leqslant{y}\leqslant{f}\left({x}\right),−\mathrm{2}\leqslant{x}\leqslant\mathrm{0}\right\} \\ $$$${is}\:{v}\:{then}\:{what}\:{the}\:{value}\:{of} \\ $$$$\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\mathrm{4}{x}\:{f}\left(\mathrm{2}{x}^{\mathrm{2}} −\mathrm{8}\right)\:{dx}\:. \\ $$$$\left({A}\right)\:\mathrm{5}{u}+\mathrm{4}{v}\:\:\:\:\:\left({D}\right)\mathrm{2}{u}+{v} \\ $$$$\left({B}\right)\:\mathrm{4}{u}+\mathrm{3}{v}\:\:\:\:\left({E}\right)\:{u}+{v} \\ $$$$\left({C}\right)\:\mathrm{3}{u}+\mathrm{2}{v} \\ $$

Answered by ajfour last updated on 15/Apr/21

u=(1/2)∫_(−6) ^( −2) f(x)dx=((F(−2)−F(−6))/2)  v=∫_(−2) ^( 0) f(x)dx=F(0)−F(−2)  I=∫_1 ^( 2) 4xf(2x^2 −8)dx  let  2x^2 −8=t  ⇒   4xdx=dt  I=∫_(−6) ^( 0) f(x)dx=F(0)−F(−6)   = 2u+v    (D) .

$${u}=\frac{\mathrm{1}}{\mathrm{2}}\int_{−\mathrm{6}} ^{\:−\mathrm{2}} {f}\left({x}\right){dx}=\frac{{F}\left(−\mathrm{2}\right)−{F}\left(−\mathrm{6}\right)}{\mathrm{2}} \\ $$$${v}=\int_{−\mathrm{2}} ^{\:\mathrm{0}} {f}\left({x}\right){dx}={F}\left(\mathrm{0}\right)−{F}\left(−\mathrm{2}\right) \\ $$$${I}=\int_{\mathrm{1}} ^{\:\mathrm{2}} \mathrm{4}{xf}\left(\mathrm{2}{x}^{\mathrm{2}} −\mathrm{8}\right){dx} \\ $$$${let}\:\:\mathrm{2}{x}^{\mathrm{2}} −\mathrm{8}={t} \\ $$$$\Rightarrow\:\:\:\mathrm{4}{xdx}={dt} \\ $$$${I}=\int_{−\mathrm{6}} ^{\:\mathrm{0}} {f}\left({x}\right){dx}={F}\left(\mathrm{0}\right)−{F}\left(−\mathrm{6}\right) \\ $$$$\:=\:\mathrm{2}{u}+{v}\:\:\:\:\left({D}\right)\:. \\ $$

Answered by tepebea last updated on 16/Apr/21

mantap...

$${mantap}... \\ $$

Commented by ajfour last updated on 16/Apr/21

what does this mean in English..

$${what}\:{does}\:{this}\:{mean}\:{in}\:{English}.. \\ $$

Commented by tepebea last updated on 16/Apr/21

good job or that′s great

$${good}\:{job}\:{or}\:{that}'{s}\:{great} \\ $$

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