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Welcome to March Calculus Madness!
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Double Integration Time
(I)
∫ ∫ sin(a*x + b*y) dx dy
= ∫ -1/a * cos(a*x + b*y) + C1 dy
= -1/(a*b) * sin(a*x + b*y) + C1*y + C2
(II)
∫ ∫ sin(a*x + b*y) dy dx
= ∫ -1/b * cos(a*x + b*y)+ C1 dx
= -1/(a*b) * sin(a*x + b*y) + C1 * x + C2
For (I) and (II) to equal, x = y
(III)
∫ ∫ e^(a*x + b*y) dx dy
= ∫ 1/a * e^(a*x + b*y)+ C1 dy
= 1/(a*b) * e^(a*x + b*y) + C1 * y + C2
(IV)
∫ ∫ e^(a*x + b*y) dy dx
= ∫ 1/b * e^(a*x + b*y) + C1 dx
= 1/(a*b) * e^(a*x + b*y) + C1 * x + C2
For (III) and (IV) to be equal, x = y
Eddie
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