Math · Calculus · Concept
Accumulation and the Fundamental Theorem
A running integral connects area accumulated so far with the rate being accumulated.
Move only the upper endpoint
Keep a fixed and define F(x) as signed area from a to x. Changing x adds or removes a thin strip. Negative f means accumulation decreases as x increases.
Why the slope equals the height
The difference F(x+h)−F(x) is the integral over the short interval from x to x+h. Dividing by h gives the average height there. At a point where f is continuous, that average tends to f(x). The running area therefore has derivative f.
Why endpoint subtraction works
If G′=f on an interval and f is continuous, F and G have the same derivative. Their difference is constant. Since F(a)=0, F(b)=G(b)−G(a), turning a limiting sum into an endpoint calculation.
Common mistakes
- F(x) is accumulated area, while F′(x) is the current height.
- Continuity conditions matter; a jump in f need not give a derivative of F at that point.
Work through an example
Find F(x)=∫₀ˣ(t−1)dt and explain when it decreases.
Accumulate a changing signed rate →Sources and scope
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