Ep 18: The Theorem That Tied All of Calculus Together | The Fundamental Theorem

Calculus has two halves — derivatives and integrals — and at first glance they look like completely different problems. Derivatives are about slopes. Integrals are about areas. The Fundamental Theorem of Calculus is the single identity that says: they are inverses of each other. That insight, which Isaac Barrow saw in a geometric form, Newton and Leibniz wrote down algebraically, and Cauchy and Riemann finally made rigorous, is the reason we treat differential and integral calculus as one subject instead of two. This video walks through both halves of the FTC, the priority dispute between Newton and Leibniz that shaped how it was published, and the 19th-century work that put the theorem on the foundations it has today. Then four fully worked examples: a polynomial integral computed with FTC Part 2, a derivative of an integral with a variable upper limit using FTC Part 1, the area under one bump of the sine curve, and the integral of one over x from 1 to e that ties area under a hyperbola to the very definition of the number e. 📚 Continues in: → 'u-Substitution' — the FTC, in action. 🎓 Best for: Calculus 1, 2, and 3 students, AP Calculus AB / BC, anyone reviewing before an exam, and self-learners who want a single coherent course instead of a pile of disconnected YouTube videos. — QED Realized — Clean explanations of the ideas that take a real lecture to land. 🔔 Subscribe so you don't miss the next one:    / @qedrealized   #Calculus #Math #Mathematics #STEM #FTC #FundamentalTheoremOfCalculus #Calculus2 #APCalculus — Tags (for search) — fundamental theorem of calculus, FTC explained, FTC part 1, FTC part 2, fundamental theorem of calculus proof, fundamental theorem of calculus examples, definite integral, antiderivative, Newton Leibniz priority dispute, Isaac Barrow, Cauchy integral, Riemann integral, calculus 1, calculus 2, AP Calculus AB, AP Calculus BC, integral calculus, derivative and integral, why calculus works, MIT calculus ⏱ Chapters 0:00 The Fundamental Theorem of Calculus 0:31 Two ideas that look unrelated 1:20 Isaac Barrow sees it first (geometrically) 2:16 Newton and Leibniz make it explicit 3:54 Cauchy and Riemann make the integral rigorous 5:04 FTC Part 1: differentiating an integral 6:10 FTC Part 2: evaluating an integral 6:50 Part 1 and Part 2 are two faces of the same theorem 7:19 A sketch of why Part 1 is true 8:42 Worked example 1: integral of x squared from 0 to 2 9:33 Sanity check against Riemann sums 10:13 Worked example 2: derivative of an integral 10:59 Why the FTC needs continuity 12:25 Worked example 3: a sine integral 13:19 Worked example 4: integral of 1/x from 1 to e 14:14 Why this matters 15:27 Recap 16:27 Thanks for watching