Calculus
Fundamental theorem of calculus
Differentiation and integration are inverses — the link that ties everything together.
Differentiation and integration are inverses — the deepest result in calculus.
Drag the right edge
y = 2·xarea [0, 2] ≈ 4
Why it's 'fundamental'
Before this theorem, finding area meant grinding through Riemann sums by hand. The FTC says: don't bother — just find an antiderivative and subtract. It welds the two halves of calculus (slopes and areas) into one tool. Drag the edge above: the area under y = 2x out to x equals x², which is exactly an antiderivative of 2x.
Part 2 (evaluation)
where F is any antiderivative of f. The accumulated area is just the change in the antiderivative.
Part 1 (the area function)
Differentiating an accumulated-area function gives back the original integrand — differentiation undoes integration.
Your turn
Use the FTC: ∫₁⁴ 2x dx = ?
Two parts
- Part 1: d/dx [∫ₐˣ f(t) dt] = f(x).
- Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a), where F' = f.