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Question Number 128282 by BHOOPENDRA last updated on 06/Jan/21

Answered by mr W last updated on 06/Jan/21

ellipse (x^2 /2^2 )+(y^2 /3^2 )=1 or x=2 cos t, y=3 sin t with 0≤t≤2π c(t)=(2 cos t, 3 sin t) c′(t)=(−2 sin t, 3 cos t) F(x,y)=(x−3y, y−2x) F(c(t))=F(2 cos t, 3 sin t) =(2 cos t−9 sin t, −4 cos t+3 sin t) ∮_C F∙ds=∫_0 ^(2π) F(c(t))∙c′(t) dt =∫_0 ^(2π) (2 cos t−9 sin t, −4 cos t+3 sin t)∙(−2 sin t, 3 cos t) dt =∫_0 ^(2π) (−4 sin t cos t+18 sin^2 t−12 cos^2 t+9 sin t cos t) dt =∫_0 ^(2π) (5 sin t cos t+18−30 cos^2 t) dt =∫_0 ^(2π) ((5/2) sin 2t+3−15 cos 2t) dt =3×2π =6π

$${ellipse}\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{\mathrm{3}^{\mathrm{2}} }=\mathrm{1} \\ $$$${or} \\ $$$${x}=\mathrm{2}\:\mathrm{cos}\:{t},\:{y}=\mathrm{3}\:\mathrm{sin}\:{t}\:{with}\:\mathrm{0}\leqslant{t}\leqslant\mathrm{2}\pi \\ $$$${c}\left({t}\right)=\left(\mathrm{2}\:\mathrm{cos}\:{t},\:\mathrm{3}\:\mathrm{sin}\:{t}\right) \\ $$$${c}'\left({t}\right)=\left(−\mathrm{2}\:\mathrm{sin}\:{t},\:\mathrm{3}\:\mathrm{cos}\:{t}\right) \\ $$$${F}\left({x},{y}\right)=\left({x}−\mathrm{3}{y},\:{y}−\mathrm{2}{x}\right) \\ $$$${F}\left({c}\left({t}\right)\right)={F}\left(\mathrm{2}\:\mathrm{cos}\:{t},\:\mathrm{3}\:\mathrm{sin}\:{t}\right) \\ $$$$=\left(\mathrm{2}\:\mathrm{cos}\:{t}−\mathrm{9}\:\mathrm{sin}\:{t},\:−\mathrm{4}\:\mathrm{cos}\:{t}+\mathrm{3}\:\mathrm{sin}\:{t}\right) \\ $$$$\oint_{{C}} {F}\centerdot{ds}=\int_{\mathrm{0}} ^{\mathrm{2}\pi} {F}\left({c}\left({t}\right)\right)\centerdot{c}'\left({t}\right)\:{dt} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \left(\mathrm{2}\:\mathrm{cos}\:{t}−\mathrm{9}\:\mathrm{sin}\:{t},\:−\mathrm{4}\:\mathrm{cos}\:{t}+\mathrm{3}\:\mathrm{sin}\:{t}\right)\centerdot\left(−\mathrm{2}\:\mathrm{sin}\:{t},\:\mathrm{3}\:\mathrm{cos}\:{t}\right)\:{dt} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \left(−\mathrm{4}\:\mathrm{sin}\:{t}\:\mathrm{cos}\:{t}+\mathrm{18}\:\mathrm{sin}^{\mathrm{2}} \:{t}−\mathrm{12}\:\mathrm{cos}^{\mathrm{2}} \:{t}+\mathrm{9}\:\mathrm{sin}\:{t}\:\mathrm{cos}\:{t}\right)\:{dt} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \left(\mathrm{5}\:\mathrm{sin}\:{t}\:\mathrm{cos}\:{t}+\mathrm{18}−\mathrm{30}\:\mathrm{cos}^{\mathrm{2}} \:{t}\right)\:{dt} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \left(\frac{\mathrm{5}}{\mathrm{2}}\:\mathrm{sin}\:\mathrm{2}{t}+\mathrm{3}−\mathrm{15}\:\mathrm{cos}\:\mathrm{2}{t}\right)\:{dt} \\ $$$$=\mathrm{3}×\mathrm{2}\pi \\ $$$$=\mathrm{6}\pi \\ $$

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