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It links differentiation and integration concepts.
F′(x)=f(x)F'(x) = f(x)F′(x)=f(x) if F(x)=∫axf(t)dtF(x) = \int_a^x f(t) dtF(x)=∫axf(t)dt.
Use F(b)−F(a)F(b) - F(a)F(b)−F(a) for an antiderivative F(x)F(x)F(x) of f(x)f(x)f(x).
Integration can be undone by differentiation.
An antiderivative F(x)F(x)F(x) satisfies F′(x)=f(x)F'(x) = f(x)F′(x)=f(x).
F(x)F(x)F(x) represents the accumulated area from aaa to xxx.
Result: 63, from [x3][x^3][x3] evaluated at 4 and 1.
The function f(x)f(x)f(x) must be continuous on [a,b][a, b][a,b].
The area between f(x)f(x)f(x) and the xxx-axis from aaa to bbb.
It profoundly impacts mathematics, physics, and engineering.
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