Question Number 103773 by bemath last updated on 17/Jul/20

∫_c ((x^2 +2xy^2 )dx+(x^2 y^2 −1)dy)  where C is the boundary of  region define by y^2 = 4x and y  =1 ?

Answered by bramlex last updated on 17/Jul/20

note that C is a closed curve .  observe that y^2 =4x intersects  x=1 when y = ± 2  Green′s Theorem yields  ∫_c ((x^2 +2xy^2 )dx+(x^2 y^2 −1)dy)=  ∫_(−2) ^2  ∫_(y^2 /4) ^1 (2xy^2 −4xy) dxdy   =∫_(−2) ^2 {(x^2 y^2 −2x^2 y)}∣_(y^2 /4) ^1  dy  = 2∫_0 ^2 (y^2 −(y^6 /(16))) dy  [ even  function]  = [(2/3)y^3 −(y^7 /(112)) ]_9 ^2 = ((16)/3) −((128)/(112))  =((16)/3)−(8/7) = ((88)/(21))