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Sagot :
The area around the given curve according to the green theorem is [tex]F. dr = 0[/tex].
According to the statement
we have to find the area enclosed by the simple closed curve that encloses the origin.
So, We know that the
The given equation is
[tex]f(x,y) = \frac{2xyi + (y^{2} - x^{2} ) j}{(x^{2} + y^{2} )^{2} }[/tex]
and
If function is in form of,
[tex]F = Pi + Qj[/tex]
and C is any positively oriented simple closed curve that encloses the origin.
Then,by use of Green's theorem
Do the partial differentiation of the given function
Then
[tex]\frac{dQ}{dx} = \frac{2x^{3} - 6xy^{2}}{(x^{2} + y^{2} )^{3}}[/tex]
and
[tex]\frac{dP}{dy} = \frac{2x^{3} - 6xy^{2}}{(x^{2} + y^{2} )^{3}}[/tex]
On substitution in Green's theorem,
We get the value
[tex]F. dr = 0[/tex]
From this it is clear that the area around the given curve is zero.
So, The area around the given curve according to the green theorem is [tex]F. dr = 0[/tex].
Learn more about Green theorem here
https://brainly.com/question/23265902
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