- #1
gtfitzpatrick
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Homework Statement
Cis the boundary of the region given by curves [itex]y = x^{2}[/itex] and y=x use Greens theorem to evaluate the following line integrals.
a) [itex]\oint(6xy-y^2 )dx[/itex]
b) [itex]\oint(6xy-y^2 )dy[/itex]
Homework Equations
The Attempt at a Solution
so greens theorem states [itex]\oint Mdx + Ndy = \int\int \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} dxdy[/itex]
so for (a) M=6xy-[itex]y^2[/itex] N=0
The 2 curves intersect at (1,1)
so [itex]\oint (6xy-y^2)dx = \int^{1}_{0} \int^{x^2}_{x} (6x-2y) dydx[/itex]
=[itex]\int^{1}_{0} -x^4 + 6x^3 -7x^2 dx[/itex]
=[itex]\frac{-31}{30}[/itex]
and for (b) N=6xy-[itex]y^2[/itex] M=0
The 2 curves intersect at (1,1)
so [itex]\oint (6xy-y^2)dy = \int^{1}_{0} \int^{x^2}_{x} (6y-2y)dydx[/itex]
=[itex]\int^{1}_{0} 2x^4 - 2x^2 dx[/itex]
=[itex]\frac{-4}{15}[/itex]
am i doing this right?