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Question Number 117493 by bobhans last updated on 12/Oct/20

$${\int }_{\mathrm{c}}(3\mathrm{xy}-{\mathrm{e}}^{\sin \mathrm{x}})\mathrm{dx}+(7\mathrm{x}+\sqrt{{\mathrm{y}}^4+1})\mathrm{dy}$$$$\mathrm{C}:\mathrm{triangle}\mathrm{with}\mathrm{vertex}(0,0),(0,1)$$$$\mathrm{and}(1,0)$$

Answered by bemath last updated on 12/Oct/20

$$\mathrm{Green}\mathrm{Theorem}$$$${\int }_{\mathrm{c}}\mathrm{Pdx}+\mathrm{Qdy}=\int {\int }_{\mathrm{D}}({\mathrm{Q}}_{\mathrm{x}}-{\mathrm{P}}_{\mathrm{y}})\mathrm{dxdy}$$$$=\int {\int }_{\mathrm{D}}(7-3\mathrm{x})\mathrm{dxdy}$$$$=\underset{0}{\stackrel{1}{\int }}\underset{0}{\stackrel{1-\mathrm{x}}{\int }}(7-3\mathrm{x})\mathrm{dydx}$$$$=\underset{0}{\stackrel{1}{\int }}(7-3\mathrm{x})(1-\mathrm{x})\mathrm{dx}$$$$={\stackrel{1}{\int }}_0(7-10\mathrm{x}+{3\mathrm{x}}^2)\mathrm{dx}$$$$={[7\mathrm{x}-{5\mathrm{x}}^2+{\mathrm{x}}^3]}_0^1=3$$

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