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

$$\mathrm{57} \\ $$

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