Derivatives = local slope stories. Integrals = accumulated area stories.
1) Derivative: slope as a local story
Intuition:
the derivative answers, “If I move a tiny bit in x, how fast does y change right here?”
The tangent line is the best straight-line story of the curve at that point.
2) Integral: area as an accumulation story
Intuition:
the integral answers, “How much total effect built up between a and b?”
It’s an accumulation of many tiny contributions f(x)·dx.