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Question Number 131810 by Fikret last updated on 08/Feb/21

f(x)= { ((−2x          ;  x≤0)),((f(x−1)   ;  x>0)) :}      ∫_0 ^(100) f(x)dx =?

$${f}\left({x}\right)=\begin{cases}{−\mathrm{2}{x}\:\:\:\:\:\:\:\:\:\:;\:\:{x}\leqslant\mathrm{0}}\\{{f}\left({x}−\mathrm{1}\right)\:\:\:;\:\:{x}>\mathrm{0}}\end{cases} \\ $$ $$ \\ $$ $$\:\:\underset{\mathrm{0}} {\overset{\mathrm{100}} {\int}}{f}\left({x}\right){dx}\:=? \\ $$

Answered by mr W last updated on 09/Feb/21

∫_0 ^(100) f(x)dx  =Σ_(k=0) ^(99) ∫_k ^(k+1) f(x)dx  =100∫_(−1) ^0 f(x)dx  =100∫_(−1) ^0 (−2x)dx  =100[−x^2 ]_(−1) ^0   =100

$$\int_{\mathrm{0}} ^{\mathrm{100}} {f}\left({x}\right){dx} \\ $$ $$=\underset{{k}=\mathrm{0}} {\overset{\mathrm{99}} {\sum}}\int_{{k}} ^{{k}+\mathrm{1}} {f}\left({x}\right){dx} \\ $$ $$=\mathrm{100}\int_{−\mathrm{1}} ^{\mathrm{0}} {f}\left({x}\right){dx} \\ $$ $$=\mathrm{100}\int_{−\mathrm{1}} ^{\mathrm{0}} \left(−\mathrm{2}{x}\right){dx} \\ $$ $$=\mathrm{100}\left[−{x}^{\mathrm{2}} \right]_{−\mathrm{1}} ^{\mathrm{0}} \\ $$ $$=\mathrm{100} \\ $$

Commented bymr W last updated on 09/Feb/21

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