The Fundamental Theorem of Calculus
What you'll learn
See why differentiation and integration are inverse operations — the theorem the whole subject turns on.
So far derivatives (slopes) and integrals (areas) look like two separate stories. The Fundamental Theorem of Calculus reveals they're the same story told backwards — and it's the result the entire subject is built on.
The area-so-far function
Let A(x) be the area under f from a fixed start a up to a moving point x. As you push x to the right, the area grows. How fast?
Nudge x by a tiny dx and you tack on a thin strip of area ≈ height × width = f(x)·dx. So the rate the area grows is exactly the height of the curve:
A′(x) = f(x)
The derivative of the area function is the original function. Accumulating (∫) and finding a rate (d/dx) undo each other.
The two parts
| Part | Statement | What it does |
|---|---|---|
| First | d/dx ∫ₐˣ f(t) dt = f(x) | differentiating an integral gives back f |
| Second | ∫ₐᵇ f(x) dx = F(b) − F(a) | evaluate any integral using an antiderivative F |
The second part is the one you compute with: to find an area, don't sum a million rectangles — find an antiderivative F (a function whose derivative is f) and subtract its values at the endpoints.
Worked example
Find ∫₀² 3x² dx. An antiderivative of 3x² is F(x) = x³ (since d/dx x³ = 3x²). Then:
∫₀² 3x² dx = F(2) − F(0) = 8 − 0 = 8.
No rectangles — just an antiderivative and a subtraction.
Why this matters
This theorem is why calculus is practical. It converts the impossible-looking limit of Riemann sums into a quick antiderivative-and-subtract. Every integral you evaluate by hand leans on it — which is why the next lesson is about finding antiderivatives.
Check your understanding
Question 1 of 2
The Fundamental Theorem of Calculus says differentiation and integration are: