Differentiation answers a question about slope: given a curve, what's its instantaneous rate of change at any point? Integration answers a question about area: given a curve, what's the space between it and the axis over some interval? For decades before calculus was fully formalised, mathematicians treated these as separate problems, solved with separate techniques. The Fundamental Theorem of Calculus proved they weren't separate at all — differentiation and integration are inverse operations of exactly the same underlying process, and once you see that, entire categories of previously hard area-and-accumulation problems become solvable by working backward from a derivative.
Area as a function, and its derivative
The theorem's first part shows that if you define a function as the running area under a curve up to some point, the derivative of that area function is simply the original curve's height at that point. That's a genuinely surprising claim on its face — accumulating area sounds like it should behave completely differently from measuring a slope — but it follows directly from thinking carefully about what happens as you nudge the endpoint of the area calculation by a tiny amount: the area added is approximately the curve's height there times that tiny width, and dividing by that tiny width to get a rate of change leaves you with the height itself, which is the definition of a derivative.
Antiderivatives turn area into arithmetic
The theorem's second, more practically powerful part follows from the first: since the area function and the original function are linked through differentiation, finding an exact area under a curve no longer requires the laborious geometric approximation techniques (like breaking a region into ever-thinner rectangles) that mathematicians relied on before calculus. Instead, you find any antiderivative of the function — something whose derivative gives back the original curve — evaluate it at the two endpoints of the interval, and subtract. What used to require a limiting process of infinitely many approximations collapses into a handful of algebraic steps, which is exactly why the theorem is considered the hinge connecting the differential and integral halves of calculus into a single coherent subject rather than two separate ones that happen to share a name.
What we're still unsure about
The Fundamental Theorem of Calculus itself is a completely settled, rigorously proven result, not something in any mathematical dispute. What's more a matter of pedagogy than open mathematics is how best to build genuine intuition for why slope and accumulated area are connected in the first place — many students can apply the theorem's mechanical steps correctly for years without the "why" ever fully clicking, and different textbooks and instructors continue to experiment with which explanation, visualisation, or historical framing actually makes the connection feel inevitable rather than just memorised.
This sits inside Definite Integrals & the Fundamental Theorem, one of eight topics in Calculus, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.