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Question Number 103607 by bemath last updated on 16/Jul/20
whatisthevalueof∫c(x+2y)dx+(4−2x)dyaroundtheellipseC:x216+y28=1inthecounterclockwisedirection?
Answered by bobhans last updated on 16/Jul/20
sincetheellipseisaclosedcurve,withGreen′sTheorem.∫C((x+2y)dx+(4−2x)dy)=∫∫R(∂∂x(4−2x)−∂∂y(x+2y))dA=∫∫R(−2−2)dA=−4∫∫RdAbecausetheareaofanellipsewithequationx216+y28=1equaltoπ.4.8=8π2weget−4∫∫R=−4×8π2=−32π2weconclude∫C((x+2y)dx+(4−2x)dy)=−32π2
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