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Leibnitz Integral [rule] (feddit.de)

submitted 2 years ago by theFibonacciEffect@feddit.de to c/196@lemmy.blahaj.zone

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[–] bh11235 15 points 2 years ago* (last edited 2 years ago) (1 children)

f is a real function from the xy plane to the reals
draw a horizontal-ish band (that is, two lines) creating your favorite wobbly band shape
the integral of f along a single vertical line within the band is itself a function of the line's x coordinate. Call the function g(x)
What is dg/dx in terms of df/dx? lf the band were perfectly horizontal, it would just be the integral along the vertical line of df/dx within the band. But it's not, so you add a term to account for the band lines moving
all this assuming f and the band are pretty enough (this is 80% of the theorem statement)

[–] lgstarn@kbin.social 2 points 2 years ago

For number 4 (or maybe a new number), if a(x) and b(x) are constant, the derivatives d/dx a(x) and d/dx b(x) are zero, so all that is left is that d/dx becomes a partial derivative and can move inside the integral.