The Fundamental Theorem of Calculus (FTC) is the bridge between derivatives and integrals — and one of the most tested concepts on the AP Calculus AB & BC exams. It shows up in both multiple-choice and free-response questions, often paired with applications like area, accumulation, and motion problems.
In this RevisionDojo guide, you’ll learn:
- The two parts of the FTC and what they mean
- How to apply them to different types of problems
- AP-style examples to practice
- Common mistakes and exam strategies
📚 The Two Parts of the Fundamental Theorem of Calculus
Part 1 – Derivatives of Integrals
If
F(x)=∫axf(t) dtF(x) = \int_a^x f(t) \, dt
then
F′(x)=f(x)F'(x) = f(x)
This means differentiating an integral with a variable upper limit gives the original function back.
Example:
ddx∫0xcos(t) dt=cos(x)\frac{d}{dx} \int_0^x \cos(t) \, dt = \cos(x)
Part 2 – Evaluating Definite Integrals
If ff is continuous on [a,b][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 lets you evaluate definite integrals using antiderivatives.
Example:
∫13(2x) dx=[x2]13=9−1=8\int_1^3 (2x) \, dx = [x^2]_1^3 = 9 - 1 = 8
🔍 AP-Style Applications
- Variable Limits
If the upper limit is g(x)g(x), apply the chain rule:
ddx∫ag(x)f(t) dt=f(g(x))⋅g′(x)\frac{d}{dx} \int_a^{g(x)} f(t) \, dt = f(g(x)) \cdot g'(x)
- Reversing Limits
∫baf(x) dx=−∫abf(x) dx\int_b^a f(x) \, dx = -\int_a^b f(x) \, dx
- Piecewise Functions
Break the integral into parts where the function changes form.
📝 AP-Style Example Problem
Problem:
Let
F(x)=∫2x21+t3 dtF(x) = \int_2^{x^2} \sqrt{1 + t^3} \, dt
Find F′(x)F'(x).
Solution:
- Use FTC Part 1 with chain rule:
F′(x)=1+(x2)3⋅ddx(x2)F'(x) = \sqrt{1 + (x^2)^3} \cdot \frac{d}{dx}(x^2)F′(x)=1+x6⋅2xF'(x) = \sqrt{1 + x^6} \cdot 2x
⚠️ Common Mistakes to Avoid
- Forgetting the chain rule when the limit is not just xx
- Mixing up Part 1 and Part 2 definitions
- Dropping the negative sign when switching integration limits
- Trying to differentiate before applying FTC correctly
📊 Practice Strategy from RevisionDojo
- Drill problems that mix FTC with substitution for more exam-like difficulty
- Practice with integrals that have functions as limits
- Combine FTC questions with motion problems (position, velocity, acceleration)
🧭 Final Advice from RevisionDojo
The FTC is a guaranteed topic on the AP Calculus exam. If you master both parts, apply the chain rule correctly, and handle variable limits confidently, you can earn quick and easy points on exam day.
