Question Number 215390 by MathematicalUser2357 last updated on 05/Jan/25
![∮_γ x^2 dx [γ: x^2 +y^2 =1]](https://www.tinkutara.com/question/Q215390.png)
$$\oint_{\gamma} {x}^{\mathrm{2}} {dx}\:\left[\gamma:\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{1}\right] \\ $$
Answered by MrGaster last updated on 05/Jan/25
![solve:∫_0 ^(2π) (cos t)^2 (−sin t)dt[∵x=cos t,y=sin t,dx=−sin t dt] − ∫_(0 ) ^(2π) cos^2 t sin t dt =−[((cos^3 t)/3)]_0 ^(2π) =−(((cos^3 (2π))/3)−((cos^3 (0))/3)) =−((1/3)−(1/3)) =0](https://www.tinkutara.com/question/Q215391.png)
$$\mathrm{solve}:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \left(\mathrm{cos}\:{t}\right)^{\mathrm{2}} \left(−\mathrm{sin}\:{t}\right){dt}\left[\because{x}=\mathrm{cos}\:{t},{y}=\mathrm{sin}\:{t},{dx}=−\mathrm{sin}\:{t}\:{dt}\right] \\ $$$$\:\:\:−\:\:\int_{\mathrm{0}\:} ^{\mathrm{2}\pi} \mathrm{cos}^{\mathrm{2}} {t}\:\mathrm{sin}\:{t}\:{dt} \\ $$$$=−\left[\frac{\mathrm{cos}^{\mathrm{3}} {t}}{\mathrm{3}}\right]_{\mathrm{0}} ^{\mathrm{2}\pi} \\ $$$$=−\left(\frac{\mathrm{cos}^{\mathrm{3}} \left(\mathrm{2}\pi\right)}{\mathrm{3}}−\frac{\mathrm{cos}^{\mathrm{3}} \left(\mathrm{0}\right)}{\mathrm{3}}\right) \\ $$$$=−\left(\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{3}}\right) \\ $$$$=\mathrm{0} \\ $$
Answered by TonyCWX08 last updated on 05/Jan/25
$$\gamma={circle}\:{with}\:{r}=\mathrm{1} \\ $$$${parametric}\:{form}: \\ $$$${x}={cos}\theta,\:{y}={sin}\theta\:{where}\:−\frac{\pi}{\mathrm{2}}\leqslant\theta\leqslant\frac{\pi}{\mathrm{2}} \\ $$$$\frac{{dx}}{{d}\theta}=−{sin}\theta,\:\frac{{dy}}{{d}\theta}={cos}\theta \\ $$$$ \\ $$$${The}\:{integral}\:{is}\:{now} \\ $$$$\underset{−\frac{\pi}{\mathrm{2}}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\left({cos}\theta\right)^{\mathrm{2}} \sqrt{\left(−{sin}\theta\right)^{\mathrm{2}} +\left({cos}\theta\right)^{\mathrm{2}} }{d}\theta \\ $$$$=\underset{−\frac{\pi}{\mathrm{2}}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}{cos}^{\mathrm{2}} \left(\theta\right)\sqrt{{sin}^{\mathrm{2}} \left(\theta\right)+{cos}^{\mathrm{2}} \left(\theta\right)}{d}\theta \\ $$$$=\underset{−\frac{\pi}{\mathrm{2}}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}{cos}^{\mathrm{2}} \left(\theta\right){d}\theta \\ $$$$=\underset{−\frac{\pi}{\mathrm{2}}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}{cos}^{\mathrm{2}} \left(\theta\right){d}\theta \\ $$$$=\left(\frac{\mathrm{1}}{\mathrm{2}}\theta+\frac{{sin}\left(\mathrm{2}\theta\right)}{\mathrm{4}}\underset{−\frac{\pi}{\mathrm{2}}} {\overset{\frac{\pi}{\mathrm{2}}} {\right)}} \\ $$$$=\frac{\pi}{\mathrm{2}} \\ $$
Commented by mr W last updated on 05/Jan/25
$${the}\:{question}\:{requests}\:…{dx},\:{not}\:…{dl}. \\ $$$${besides}\:{for}\:{the}\:{closed}\:{path}\:{it}\:{should} \\ $$$${be}\:−\pi\leqslant\theta\leqslant\pi,\:{not}\:−\frac{\pi}{\mathrm{2}}\leqslant\theta\leqslant\frac{\pi}{\mathrm{2}}. \\ $$