Line Integral around a Square
Evaluate the following line integral: where is the path around the square with vertices and .
The answer is 48.
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Using Green's Theorem, Q is in terms of dy and P is in terms of dx, So Q = 6x^2 + 5x + 3y And P = 4x^2 + 3x + 5y Green's Theorem states that ∫ {c} Pdx + Qdy = ∫∫ (∂Q/∂x - ∂P/∂y)dA So ∫ {c} [(4x^2+3x+5y)dx + (6x^2+5x+3y)dy] = ∫∫ [(12x+5) - 5)]dA , x from 0 to 2, y from 0 to 2.