Wednesday, March 16, 2022

March Calculus Madness Sweet Sixteen - Day 1: Double Integration

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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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