Asked by sara
Use Green's Theorem to evaluate the line integral along the given positively oriented curve.
integral of xy2 dx + 4x2y dy
C is the triangle with vertices (0, 0), (2, 2), and (2, 4)
integral of xy2 dx + 4x2y dy
C is the triangle with vertices (0, 0), (2, 2), and (2, 4)
Answers
Answered by
Steve
Green's Theorem says that
∮<sub>C</sub>Mdx+Ndy = ∫∫<sub>R</sub>Mx+Ny dx dy
Nx = 8xy
My = 2xy
So, we end up with
∫[0,2]∫[x,2x] 10xy dy dx
= ∫[0,2] 5xy^2 [x,2x] dx
= ∫[0,2] 5x(4x^2-x^2) dx
= ∫[0,2] 15x^3 dx
= 5x^4 [0,2]
= 80
∮<sub>C</sub>Mdx+Ndy = ∫∫<sub>R</sub>Mx+Ny dx dy
Nx = 8xy
My = 2xy
So, we end up with
∫[0,2]∫[x,2x] 10xy dy dx
= ∫[0,2] 5xy^2 [x,2x] dx
= ∫[0,2] 5x(4x^2-x^2) dx
= ∫[0,2] 15x^3 dx
= 5x^4 [0,2]
= 80
Answered by
Steve
Oops. That is Nx - My
I'll let you make the change and redo the arithmetic.
I'll let you make the change and redo the arithmetic.
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