Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

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

$$\mathrm{If}\underset{\mathrm{a}}{\stackrel{\mathrm{b}}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\underset{\mathrm{a}}{\stackrel{\mathrm{b}}{\int }}\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx},\mathrm{we}\mathrm{does}\mathrm{not}\mathrm{claim}\mathrm{that}$$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{g}\left(\mathrm{x}\right).\mathrm{but}\underset{\mathrm{a}}{\stackrel{\mathrm{b}}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\underset{\mathrm{a}}{\stackrel{\mathrm{b}}{\int }}\mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{x})\mathrm{dx}$$$$\mathrm{then}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{x}).$$

Answered by talminator2856791 last updated on 05/Feb/21

$$\mathrm{false}$$

Answered by talminator2856791 last updated on 05/Feb/21

$$\mathrm{take}\mathrm{any}\mathrm{two}\mathrm{straight}\mathrm{lines}f\left(x\right),g\left(x\right)\mathrm{of}\mathrm{different}\mathrm{gradients}.$$$$\mathrm{from}\mathrm{point}\mathrm{of}\mathrm{intersection}(m;n),\mathrm{take}\mathrm{any}\mathrm{real}p\mathrm{such}\mathrm{that}$$$$\mid f\left(p\right)\mid ,\mid g\left(p\right)\mid ⩽\mid n\mid ,\mathrm{then}$$$${\int }_{m-p}^{m+p}f\left(x\right)dx={\int }_{m-p}^{m+p}g\left(x\right)dx$$$$\Rightarrow {\int }_a^{b}f\left(x\right)dx={\int }_a^{b}g\left(x\right)dx$$$$\mathrm{the}\mathrm{assertion}\mathrm{that}f\left(x\right)=g\left(x\right)\mathrm{is}\mathrm{false},\mathrm{as}f\left(x\right)\mathrm{and}g\left(x\right)$$$$\mathrm{can}\mathrm{not}\mathrm{be}\mathrm{the}\mathrm{same}\mathrm{line}\mathrm{if}\mathrm{their}\mathrm{gradients}\mathrm{are}\mathrm{different}.$$$$$$

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

$$c=\frac{{\int }_a^{b}f\left(x\right)dx}{b-a}$$$$g\left(x\right)=c$$$$f\left(x\right)=anyfunction$$$$f\left(x\right)\ne c$$

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

$$yes,ithink.$$$$if{\int }_a^{b}f\left(x\right)dx={\int }_a^{b}g\left(x\right)dxforanya,b\in R$$$$\Rightarrow \underset{b\to a}{\lim }{\int }_a^{b}f\left(x\right)dx=\underset{b\to a}{\lim }{\int }_a^{b}g\left(x\right)dx$$$$\Rightarrow \underset{b\to a}{\lim }f\left(a\right)(b-a)=\underset{b\to a}{\lim }g\left(a\right)(b-a)$$$$\Rightarrow f\left(a\right)=g\left(a\right)$$$$\Rightarrow f\left(x\right)=g\left(x\right)$$

Commented by mr W last updated on 05/Feb/21

$${\int }_a^{b}f\left(x\right)dx=redshadedarea$$$${\int }_a^{b}g\left(x\right)dx=blueshadedarea$$$${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}g\left(x\right)dxmeansonly$$$$thattheredshadedareaandthe$$$$blueshadedareaareequal.it{doesn}^{\prime }t$$$$meanthatf\left(x\right)=g\left(x\right)!$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com