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Welcome to March Calculus Madness!
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For the two-variable function f(x,y), can we assume that ∫ ∫ f(x,y) dx dy = ∫∫ f(x,y) dy dx?
Two simple examples:
Equation 1:
∫ ∫ x^2 + y^2 dx dy
= ∫ x^3/3 + y^2 ∙ x + C1 dy
x^3 ∙ y/3 + y^3 ∙ x/3 + C1 ∙ y + C2
Equation 2:
∫ ∫ x^2 + y^2 dy dx
∫ x^2 ∙ y + y^3/3 + C1 dx
x^3 ∙ y/3 + y^3 ∙ x/3 + C1∙ x + C2
However, for both Equation 1 and Equation 2 to be equal:
x^3 ∙ y/3 + y^3 ∙ x/3 + C1 ∙ y + C2 = x^3 ∙ y/3 + y^3 ∙ x/3 + C1∙ x + C2
C1 ∙ y = C1 ∙ x
y = x
By this example alone, we cannot assume that ∫ ∫ f(x,y) dx dy = ∫∫ f(x,y) dy dx.
Eddie
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