Integral function and the fundamental theorem
Balance areas up to x, find extrema of the integral function and calculate with J(x) = F(x) − F(a).
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Guide
The integral function J_a(x) = ∫ₐˣ f(t) dt gives the signed area balance between a and x: areas above the x-axis count positive, below negative. The geometric area is always positive.
Fundamental theorem: J_a′(x) = f(x). Where f is positive, J grows; where f is negative, J falls. If f changes sign from + to −, J has a maximum there; if f has an extremum, J has an inflection point.
With an antiderivative F: J_a(x) = F(x) − F(a). In a word problem, if f is the inflow rate, J is the change of the filled amount; the initial amount is added.