[1 Polnts] DETAIS MY NOTES SCALCET9 16.4.021. Use Green's theorem to find the work done by the force {:F(x,y)=x(x+y)+x^(2)) in moving a particle from the origin along the x-a ws to (5,0), then along the line segment to (0,5), and then beac to the ongin along the r-wils. ◻ Shre Aresen

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[1 Polnts] DETAIS MY NOTES SCALCET9 16.4.021. Use Green's theorem to find the work done by the force  {:F(x,y)=x(x+y)+x^(2)) in moving a particle from the origin along the  x-a ws to  (5,0), then along the line segment to  (0,5), and then beac to the ongin along the  r-wils.  ◻ Shre Aresen

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Set up problem using Green's theorem: Understand and set up the problem using Green's theorem. Green's theorem relates a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. We need to convert the force field into a form suitable for applying Green's theorem. F(x, y) = x(x + y) + x^2 can be rewritten as P(x, y) = x(x + y) and Q(x, y) = x^2.Apply Green's theorem: Apply Green's theorem. Green's theorem states that the line integral of F around C is equal to the double integral over D of (∂Q/∂x - ∂P/∂y) dA. Calculate ∂Q/∂x and ∂P/∂y: ∂Q/∂x = 2x, ∂P/∂y = x. So, ∂Q/∂x - ∂P/∂y = 2x - x = x.Set up double integral: Set up the double integral. The region D is a right triangle with vertices at (0,0), (5,0), and (0,5). The limits of integration are from x = 0 to x = 5 and for each x, y goes from 0 to 5 - x. Set up the integral: ∫ (from x=0 to 5) ∫ (from y=0 to 5-x) x dy dx.Calculate double integral: Calculate the double integral. First, integrate with respect to y: ∫ (from y=0 to 5-x) x dy = x[y] from 0 to 5-x = x(5-x). Now, integrate with respect to x: ∫ (from x=0 to 5) x(5-x) dx = ∫ (from x=0 to 5) (5x - x^2) dx.Finish calculation: Finish the calculation. Calculate ∫ (from x=0 to 5) (5x - x^2) dx: = [5x^2/2 - x^3/3] from 0 to 5 = (5*25/2 - 125/3) = (125/2 - 125/3) = (375/6 - 250/6) = 125/6.

Use Green's theorem to find the work done by the force $\mathbf{F}(x,y)=x(x+y)+x^{2})$ in moving a particle from the origin along the $x$ -axis to $(5,0)$ , then along the line segment to $(0,5)$ , and then back to the origin along the $y$ -axis.

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Use Green's theorem to find the work done by the force F(x,y)=x(x+y)+x2)\mathbf{F}(x,y)=x(x+y)+x^{2})F(x,y)=x(x+y)+x2) in moving a particle from the origin along the xxx-axis to (5,0)(5,0)(5,0), then along the line segment to (0,5)(0,5)(0,5), and then back to the origin along the yyy-axis.

Use Green's theorem to find the work done by the force $\mathbf{F}(x,y)=x(x+y)+x^{2})$ in moving a particle from the origin along the $x$ -axis to $(5,0)$ , then along the line segment to $(0,5)$ , and then back to the origin along the $y$ -axis.