The Leibniz calc notation
\int f dg
is a generalization of f * Delta g (duh):

distance = rate * time -> d = \int r dt
Actually, this is much clearer if you think of time as:
- the independent
- AFFINE
variable:
it's (rate * DELTA t)
(time -elapsed- is a difference)

…so it's much clearer why it's a -dt-
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Actually, this is rather subtler:
- the function you're integrating must be -vector-valued-
  (can have affine-valued etc.)
- ...and actually, the base needn't even be affine:
  you just need the measure
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