Question Number 195046 by Mathstar last updated on 22/Jul/23
Commented by Mathstar last updated on 23/Jul/23
$$ \\ $$$$\:\:\mathrm{Given}\:\mathrm{curve}\:\mathrm{ysin}\left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{a}\:\mathrm{green}\: \\ $$$$\:\:\:\mathrm{square};\:\mathrm{solve}\:\mathrm{for}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}.\:\: \\ $$
Answered by a.lgnaoui last updated on 23/Jul/23
$$\mathrm{aire}\:\left(\mathrm{partie}\:\mathrm{verte}\right)=\int_{\frac{\boldsymbol{\pi}}{\mathrm{2}}ā\mathrm{a}} ^{\frac{\boldsymbol{\pi}}{\mathrm{2}}+\mathrm{a}} \mathrm{ydx}=\mathrm{2}\boldsymbol{\mathrm{a}}\mathrm{sin}\:\boldsymbol{\mathrm{a}} \\ $$$$\boldsymbol{\mathrm{pour}}\:\:\mathrm{a}=\frac{\boldsymbol{\pi}}{\mathrm{3}}\:\:\:\:\mathrm{Aire}=\frac{\boldsymbol{\pi}\sqrt{\mathrm{3}}}{\mathrm{6}}\:\:\:\:\:\:? \\ $$
Commented by Mathstar last updated on 23/Jul/23
Thank you.
Answered by mr W last updated on 23/Jul/23
$${s}={side}\:{length}\:{of}\:{square} \\ $$$$\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}ā\frac{{s}}{\mathrm{2}}\right)={s} \\ $$$$\mathrm{cos}\:\frac{{s}}{\mathrm{2}}={s} \\ $$$$\Rightarrow{s}\approx\mathrm{0}.\mathrm{900367} \\ $$$${area}\:{of}\:{square}\:={s}^{\mathrm{2}} \approx\mathrm{0}.\mathrm{81} \\ $$
Commented by Mathstar last updated on 23/Jul/23
Commented by mr W last updated on 23/Jul/23
Commented by mr W last updated on 23/Jul/23
$${at}\:{point}\:{A}: \\ $$$${x}_{{A}} =\frac{\pi}{\mathrm{2}}ā\frac{{s}}{\mathrm{2}} \\ $$$${y}_{{A}} ={s} \\ $$$${point}\:{A}\:{is}\:{on}\:{the}\:{curce}\:{y}=\mathrm{sin}\:{x},\:{hence} \\ $$$${s}=\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}ā\frac{{s}}{\mathrm{2}}\right) \\ $$
Commented by Mathstar last updated on 23/Jul/23
Thank you so much, Mr W. It is indisputable that you are one of the best Mathematicians on this platform.