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gtfitzpatrick
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Homework Statement
Let C be the boundary of the region bounded by the curves y=[itex]x^{2}[/itex] and y=x. Assuming C is oriented counter clockwise, Use green's theorem to evaluate the following line integrals (a) [itex]\oint(6xy-y^2)dx[/itex] and (b) [itex]\oint(6xy-y^2)dy[/itex]
Homework Equations
The Attempt at a Solution
[itex]\int^{0}_{1} 6x^2 - x^2[/itex]
[itex]\int^{0}_{1} 5x^2 [/itex] = -[itex]\frac{5}{3}[/itex]
and
[itex]\int^{1}_{0} 6x^3 - x^4[/itex] = [itex]\frac{6}{4} - \frac{1}{5} = \frac{13}{10}[/itex]
so [itex]\oint[/itex] = -[itex]\frac{11}{30}[/itex]
but
[itex]\int\int_{R}[/itex] [itex]\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}[/itex])dxdy
M=6xy-[itex]y^{2}[/itex] and N=0
[itex]\frac{\partial M}{\partial y} 6x-2y[/itex]
[itex]\int\int_{R}[/itex](6x-2y)dxdy
[itex]\int^{1}_{0} [ \int^{x}_{y=x^2} (6x-2y)dy] dx [/itex]
[itex]\int^{1}_{0} 5x^2 - 6x^3 - x^4 dx [/itex]
= [itex]\frac{-1}{30}[/itex]
anyone got any idea what I am doing wrong here!stumped