Greens Theorem

Can someone show me step by step how to work this problem out please.

Let C be the triangle with vertices (0,0), (1,0), and (1,2). Find the following line integral by two methods, directly and using greens theorem.

2xy^2dx + yx^2dy

Thanks!!!

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asked by julia
  1. directly
    first from (0,0) to (1,0)
    y = 0
    x goes from 0 to 1
    both terms are zero, so no contribution
    second from (1,0) to (1,2)
    Here dx = 0, x=1 and y goes from 0 to 2
    so from 0 to 2 of y dy
    = 2^2/2 = 2 total so far
    now from (1,2) to(0,0)
    dy = 2dx
    y = 2x
    so the integrand is
    2 x (2x)^2 dx + 2x^3(2dx)
    = 8x^3 dx + 4x^3 dx
    = 12x^3 dx
    from x = 1 to x = 0
    3(0)^4 -3 (1^4) = -3
    2 - 3 = -1 finally
    NOW, using Green
    closed path integral of Ldx+Mdy = integral over surface inside of
    (dM/dx -dL/dy)
    here L = 2xy^2
    so dL/dy = 4xy
    here M = y x^2
    so dM/dx =2xy
    so
    dM/dx - dL/dy = integrand = -2xy
    integral of
    -2xy dy dx from y = 0 to y = 2x
    then from x = 0 to 1
    -2 *int x dx *int y dy
    -2 * int x dx *[ y^2/2 from y = 0 to y = 2x]

    -2 int x dx *2x^2

    -4 int x^3 from 0 to 1

    -4 (1^4/4) = -1

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    posted by Damon

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