The Fundamental Theorem

Areas from antiderivatives

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• The area function $A(x) = \int_a^x f$ satisfies $A'(x) = f(x)$.
• FTC: $\int_a^b f(x)\,dx = F(b) - F(a)$ for any antiderivative $F$.
• Derivatives and integrals undo each other.

Steps

1. Area that grows
2. Why?
3. The Fundamental Theorem of Calculus
4. The x² area, exactly
5. Between two limits
6. Area under a sine arch