References: Integration and the fundamental theorem

Source material

Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Calculus, Chapter 11: "Integration and the fundamental theorem of calculus"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/integration
  Series index: https://www.3blue1brown.com/?topic=calculus
  License: copyright Grant Sanderson; videos published on his site and YouTube
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Watch this next

• Integration and the fundamental theorem of calculus (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors. Watching the rectangles thin out into a smooth area, and then watching the fundamental theorem reveal that area as a difference of antiderivative values, is the clearest way to feel why “find a function whose rate of change is the integrand” computes an accumulation. About twenty minutes.

Going deeper

• Essence of Calculus (full series) by 3Blue1Brown. The series this track follows. The previous Clawdemy lesson made the limit precise; the next chapter (What does area have to do with slope?) unpacks geometrically why the fundamental theorem holds.

• Khan Academy: Calculus for a slower, exercise-driven treatment of Riemann sums, definite and indefinite integrals, and the fundamental theorem, with practice problems and immediate feedback.

Adjacent topics

Where this sits in the track and the wider curriculum.

• The essence of calculus (the first lesson). This lesson is the formal answer to that one. The circle-area derivation by slicing into rings was the integral ∫_0^R 2πr dr = πR^2, computed informally; the fundamental theorem now delivers it in a line, and the rate-vs-accumulation duality glimpsed there is stated precisely as “differentiation and integration are inverse operations.”

• Continuous probability (across the AI tracks). Probability densities, expected values, entropy, and KL divergence are all integrals, and the loss functions of generative models and variational methods are built from them. This lesson is the calculus prerequisite: the fundamental theorem is what links the rates a model learns to the probabilities and expectations it optimizes.