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If-f-x-x-2-x-2-f-x-3-x-gt-2-Find-1-53-f-x-dx-




Question Number 206613 by hardmath last updated on 20/Apr/24
If   f(x) =  { ((x^2    ,   x ≤ 2)),((f(x−3)   ,   x > 2)) :}  Find   ∫_(−1) ^( 53)  f(x) dx = ?
$$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\begin{cases}{\mathrm{x}^{\mathrm{2}} \:\:\:,\:\:\:\mathrm{x}\:\leqslant\:\mathrm{2}}\\{\mathrm{f}\left(\mathrm{x}−\mathrm{3}\right)\:\:\:,\:\:\:\mathrm{x}\:>\:\mathrm{2}}\end{cases} \\ $$$$\mathrm{Find}\:\:\:\int_{−\mathrm{1}} ^{\:\mathrm{53}} \:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\ $$
Answered by MM42 last updated on 20/Apr/24
57
$$\mathrm{57} \\ $$
Answered by MM42 last updated on 22/Apr/24
∀ −1≤x≤2  &   2<x≤5  & 5<x≤8  &... 50<x≤53   ⇒f(x)=x^2     ;  for  18   terms  ⇒∫^(   53) _(−1) f(x)dx=18∫_(−1) ^(  2) x^2 dx=54 ✓
$$\forall\:−\mathrm{1}\leqslant{x}\leqslant\mathrm{2}\:\:\&\:\:\:\mathrm{2}<{x}\leqslant\mathrm{5}\:\:\&\:\mathrm{5}<{x}\leqslant\mathrm{8}\:\:\&…\:\mathrm{50}<{x}\leqslant\mathrm{53}\: \\ $$$$\Rightarrow{f}\left({x}\right)={x}^{\mathrm{2}} \:\:\:\:;\:\:{for}\:\:\mathrm{18}\:\:\:{terms} \\ $$$$\Rightarrow\underset{−\mathrm{1}} {\int}^{\:\:\:\mathrm{53}} {f}\left({x}\right){dx}=\mathrm{18}\int_{−\mathrm{1}} ^{\:\:\mathrm{2}} {x}^{\mathrm{2}} {dx}=\mathrm{54}\:\checkmark \\ $$$$ \\ $$
Commented by MM42 last updated on 21/Apr/24
Commented by Tinku Tara last updated on 20/Apr/24
2+17×3=53  Total 18 terms. Not 19
$$\mathrm{2}+\mathrm{17}×\mathrm{3}=\mathrm{53} \\ $$$$\mathrm{Total}\:\mathrm{18}\:\mathrm{terms}.\:\mathrm{Not}\:\mathrm{19} \\ $$
Commented by MM42 last updated on 21/Apr/24
ok .thanks
$${ok}\:.{thanks} \\ $$
Commented by hardmath last updated on 22/Apr/24
cool dear professor thank you
$$\mathrm{cool}\:\mathrm{dear}\:\mathrm{professor}\:\mathrm{thank}\:\mathrm{you} \\ $$
Commented by MM42 last updated on 23/Apr/24
 ⋛
$$\:\underline{\underbrace{\lesseqgtr}} \\ $$

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