Introduction: Why the Fundamental Theorem of Calculus Matters
The Fundamental Theorem of Calculus (FTC) is one of the most important concepts in AP Calculus AB and BC. It bridges the two major branches of calculus — differentiation and integration — and shows how they are intimately connected.
On the AP Calculus exam, you will almost certainly encounter multiple-choice and free-response questions testing your understanding of FTC. Whether it’s evaluating definite integrals, applying antiderivatives, or interpreting accumulation functions, this theorem is central to success.
In this guide, we’ll break down both parts of the theorem, show examples, and explain strategies to avoid common mistakes. For more structured study plans and solved problems, check out RevisionDojo’s AP Calculus hub.
The Two Parts of the Fundamental Theorem of Calculus
Part 1: Derivative of an Accumulation Function
If F(x)=∫axf(t) dtF(x) = \int_a^x f(t)\,dt, then
F′(x)=f(x)F'(x) = f(x)
This means: the derivative of an integral function equals the original function inside the integral.
Example 1:
Let F(x)=∫0xcos(t) dtF(x) = \int_0^x \cos(t)\,dt.
Then F′(x)=cos(x)F'(x) = \cos(x).
This is frequently tested in multiple-choice questions where you must quickly recognize accumulation functions.
Part 2: Evaluating Definite Integrals Using Antiderivatives
If ff is continuous on [a,b], and FF is an antiderivative of ff, then:
∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)
This is the computational side of FTC — it lets you calculate areas under curves quickly using antiderivatives.
Example 2:
Evaluate ∫13(2x) dx\int_1^3 (2x)\,dx.
