# If-a-b-f-x-dx-a-b-g-x-dx-is-f-x-g-x-true-or-false-

Question Number 131505 by benjo_mathlover last updated on 05/Feb/21
$$\mathrm{If}\:\int_{\mathrm{a}} ^{\mathrm{b}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\int_{\mathrm{a}} ^{\mathrm{b}} \mathrm{g}\left(\mathrm{x}\right)\mathrm{dx} \\$$$$\mathrm{is}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{g}\left(\mathrm{x}\right)\:?\: \\$$$$\mathrm{true}\:\mathrm{or}\:\mathrm{false}? \\$$
Commented by EDWIN88 last updated on 05/Feb/21
$$\mathrm{If}\:\underset{\mathrm{a}} {\overset{\mathrm{b}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\underset{\mathrm{a}} {\overset{\mathrm{b}} {\int}}\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}\:,\:\mathrm{we}\:\mathrm{does}\:\mathrm{not}\:\mathrm{claim}\:\mathrm{that}\: \\$$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{g}\left(\mathrm{x}\right).\:\mathrm{but}\:\underset{\mathrm{a}} {\overset{\mathrm{b}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\underset{\mathrm{a}} {\overset{\mathrm{b}} {\int}}\mathrm{f}\left(\mathrm{a}+\mathrm{b}−\mathrm{x}\right)\mathrm{dx} \\$$$$\mathrm{then}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{a}+\mathrm{b}−\mathrm{x}\right). \\$$
Answered by talminator2856791 last updated on 05/Feb/21
$$\:\mathrm{false} \\$$
Answered by talminator2856791 last updated on 05/Feb/21
$$\:\mathrm{take}\:\mathrm{any}\:\mathrm{two}\:\mathrm{straight}\:\mathrm{lines}\:{f}\left({x}\right),\:{g}\left({x}\right)\:\mathrm{of}\:\mathrm{different}\:\mathrm{gradients}.\:\:\:\:\:\:\:\: \\$$$$\:\mathrm{from}\:\mathrm{point}\:\mathrm{of}\:\mathrm{intersection}\:\left({m};{n}\right),\:\mathrm{take}\:\mathrm{any}\:\mathrm{real}\:{p}\:\:\mathrm{such}\:\mathrm{that} \\$$$$\:\mid{f}\left({p}\right)\mid,\:\mid{g}\left({p}\right)\mid\:\leqslant\:\mid{n}\mid,\:\:\mathrm{then} \\$$$$\:\:\:\:\:\:\:\:\int_{{m}−{p}} ^{\:{m}+{p}} {f}\left({x}\right){dx}\:=\:\int_{{m}−{p}} ^{\:{m}+{p}} {g}\left({x}\right){dx} \\$$$$\:\Rightarrow\:\int_{{a}} ^{\:{b}} {f}\left({x}\right){dx}\:=\:\int_{{a}} ^{\:{b}} {g}\left({x}\right){dx} \\$$$$\:\mathrm{the}\:\mathrm{assertion}\:\mathrm{that}\:{f}\left({x}\right)\:=\:{g}\left({x}\right)\:\mathrm{is}\:\mathrm{false},\:\mathrm{as}\:{f}\left({x}\right)\:\mathrm{and}\:{g}\left({x}\right) \\$$$$\:\mathrm{can}\:\mathrm{not}\:\mathrm{be}\:\mathrm{the}\:\mathrm{same}\:\mathrm{line}\:\mathrm{if}\:\mathrm{their}\:\mathrm{gradients}\:\mathrm{are}\:\mathrm{different}. \\$$$$\: \\$$
Commented by benjo_mathlover last updated on 05/Feb/21
$$\mathrm{give}\:\mathrm{me}\:\mathrm{a}\:\mathrm{counterexample} \\$$
Answered by TheSupreme last updated on 05/Feb/21
$${c}=\frac{\int_{{a}} ^{{b}} {f}\left({x}\right){dx}}{{b}−{a}} \\$$$${g}\left({x}\right)={c}\: \\$$$${f}\left({x}\right)={any}\:{function} \\$$$${f}\left({x}\right)\neq{c} \\$$
Commented by talminator2856791 last updated on 05/Feb/21
$$\:\mathrm{is}\:\mathrm{this}\:\mathrm{assumption}? \\$$
Answered by mr W last updated on 05/Feb/21
Commented by prakash jain last updated on 05/Feb/21
$$\mathrm{If}\:\int_{{a}} ^{{b}} {f}\left({x}\right){dx}=\int_{{a}} ^{{b}} {g}\left({x}\right){dx} \\$$$$\mathrm{for}\:\mathrm{all}\:{a},{b}\:\in\:\mathbb{R}\:\mathrm{then}\:{f}\left({x}\right)={g}\left({x}\right) \\$$$$\mathrm{correct}? \\$$
Commented by mr W last updated on 05/Feb/21
$${yes},\:{i}\:{think}. \\$$$${if}\:\int_{{a}} ^{{b}} {f}\left({x}\right){dx}=\int_{{a}} ^{{b}} {g}\left({x}\right){dx}\:{for}\:{any}\:{a},{b}\in{R} \\$$$$\Rightarrow\underset{{b}\rightarrow{a}} {\mathrm{lim}}\int_{{a}} ^{{b}} {f}\left({x}\right){dx}=\underset{{b}\rightarrow{a}} {\mathrm{lim}}\int_{{a}} ^{{b}} {g}\left({x}\right){dx} \\$$$$\Rightarrow\underset{{b}\rightarrow{a}} {\mathrm{lim}}{f}\left({a}\right)\left({b}−{a}\right)=\underset{{b}\rightarrow{a}} {\mathrm{lim}}{g}\left({a}\right)\left({b}−{a}\right) \\$$$$\Rightarrow{f}\left({a}\right)={g}\left({a}\right) \\$$$$\Rightarrow{f}\left({x}\right)={g}\left({x}\right) \\$$
Commented by mr W last updated on 05/Feb/21
$$\int_{{a}} ^{{b}} {f}\left({x}\right){dx}={red}\:{shaded}\:{area}\: \\$$$$\int_{{a}} ^{{b}} {g}\left({x}\right){dx}={blue}\:{shaded}\:{area}\: \\$$$$\int_{{a}} ^{{b}} {f}\left({x}\right){dx}=\int_{{a}} ^{{b}} {g}\left({x}\right){dx}\:{means}\:{only} \\$$$${that}\:{the}\:{red}\:{shaded}\:{area}\:{and}\:{the} \\$$$${blue}\:{shaded}\:{area}\:{are}\:{equal}.\:{it}\:{doesn}'{t} \\$$$${mean}\:{that}\:{f}\left({x}\right)={g}\left({x}\right)! \\$$