∫Understand the Fundamental Theorem of Calculus
Build the Fundamental Theorem of Calculus from accumulation first — feel why differentiation undoes integration before you ever see the formal statement. You'll finish by pulling a real dataset and verifying the theorem numerically with your own code.
Phase 1Accumulation Functions and the Inverse Discovery
Feel accumulation functions act like running totals
Area is a function, not a single number
6 minArea is a function, not a single number
The height of f tells you how fast A is growing
7 minThe height of f tells you how fast A is growing
Try to break the pattern — it won't break
6 minTry to break the pattern — it won't break
Now read FTC and recognize it as something you already know
7 minNow read FTC and recognize it as something you already know
Phase 2Evaluating Integrals with Both Parts of FTC
Evaluate definite integrals using FTC Part 1 and 2
Part 2 is the shortcut: find an antiderivative, subtract endpoints
7 minPart 2 is the shortcut: find an antiderivative, subtract endpoints
When there's an integral inside a derivative, reach for Part 1
6 minWhen there's an integral inside a derivative, reach for Part 1
When the upper limit is a function of x, the chain rule shows up
7 minWhen the upper limit is a function of x, the chain rule shows up
The problem tells you which part of FTC to use — read it right
6 minThe problem tells you which part of FTC to use — read it right
Negative area is real — the integral knows which direction you swept
6 minNegative area is real — the integral knows which direction you swept
Phase 3FTC as the Bridge Between Area, Antiderivatives, and the Chain Rule
See FTC tie area, antiderivatives, and chain rule together
Three things you thought were separate are the same theorem
6 minThree things you thought were separate are the same theorem
When both limits are functions of x, Leibniz handles it
7 minWhen both limits are functions of x, Leibniz handles it
Physics, probability, and economics all wear the same clothes
7 minPhysics, probability, and economics all wear the same clothes
Riemann sums approximate the integral — and they expose what FTC is doing
7 minRiemann sums approximate the integral — and they expose what FTC is doing
Phase 4Your Own FTC Verification on Real Data
Verify the theorem on a real-world dataset you pick
Pull real data, accumulate it, differentiate it, and check FTC
25 minPull real data, accumulate it, differentiate it, and check FTC
Frequently asked questions
- What does the Fundamental Theorem of Calculus actually say?
- This is covered in the “Understand the Fundamental Theorem of Calculus” learning path. Start with daily 5-minute micro-lessons that build from fundamentals to hands-on application.
- What is the difference between FTC Part 1 and FTC Part 2?
- This is covered in the “Understand the Fundamental Theorem of Calculus” learning path. Start with daily 5-minute micro-lessons that build from fundamentals to hands-on application.
- Why are differentiation and integration considered inverse operations?
- This is covered in the “Understand the Fundamental Theorem of Calculus” learning path. Start with daily 5-minute micro-lessons that build from fundamentals to hands-on application.
- How does Leibniz's rule extend the Fundamental Theorem of Calculus?
- This is covered in the “Understand the Fundamental Theorem of Calculus” learning path. Start with daily 5-minute micro-lessons that build from fundamentals to hands-on application.
- How can I verify the Fundamental Theorem of Calculus numerically?
- This is covered in the “Understand the Fundamental Theorem of Calculus” learning path. Start with daily 5-minute micro-lessons that build from fundamentals to hands-on application.
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