Pure · Integration
Chapter 1 · 4
The idea
Definite integration and area under a curve
Why evaluating F(b) − F(a) gives the area under a curve, what a negative integral actually means, and why "integral" and "area" are not always the same number.
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In this lesson — start anywhere
Pure · Integration
Definite integration and area under a curve
Why evaluating F(b) − F(a) gives the area under a curve, what a negative integral actually means, and why "integral" and "area" are not always the same number.
Why it works
The area collected so far
Imagine sweeping from left to right under the curve , and let be the area collected so far. How fast is that area growing? When you move a tiny step to the right, you add a sliver that is wide and (almost exactly) tall — so you gain about of area. Gain per unit step:The fundamental theorem
Read that again: the area-so-far function is a function whose derivative is . That is exactly what an antiderivative is. So the machinery you built for reversing differentiation is, astonishingly, also the machinery for measuring area. That's why the same symbol shows up in both jobs — it's not a coincidence, it's the Fundamental Theorem of Calculus.Keep reading — free
The rest of the explanation, plus 2 worked examples you step through move by move.
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