Question Number 199122 by MathedUp last updated on 28/Oct/23
$$\mathrm{Evaluate}.\int\int_{\:\boldsymbol{\mathcal{S}}} \:\hat {\boldsymbol{\mathrm{F}}}\centerdot\mathrm{d}\hat {\boldsymbol{\mathrm{S}}} \\ $$$$\mathrm{Where}\:{f}\left({x},{y}\right)={x}^{\mathrm{2}} −\mathrm{3}{y}+\mathrm{2}\:\:, \\ $$$$\hat {\boldsymbol{\mathrm{F}}}\left({x},{y},{z}\right)=\mathrm{3}{x}\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} +\mathrm{2}{z}\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} +\left(\mathrm{1}−{y}^{\mathrm{2}} \right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \: \\ $$$$\mathrm{plane}\:\mathrm{with}\:\mathrm{vertices}\:\left(\mathrm{0},\mathrm{0}\right)\:,\:\left(\mathrm{2},\mathrm{0}\right)\:\mathrm{and}\:\left(\mathrm{2},−\mathrm{4}\right)\:\mathrm{Oriented} \\ $$$$\mathrm{in}\:\mathrm{The}\:\mathrm{negative}\:{z}−\mathrm{Axies}\:\mathrm{direction}. \\ $$