Fundamental Theorem of Calculus
Differentiation and integration are inverse operations.
About this topic
The Fundamental Theorem of Calculus connects derivatives and integrals. Part 1: the derivative of the area function ∫ₐˣ f(t) dt is f(x). Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f.
This lets us compute areas by finding antiderivatives instead of adding infinitely many rectangles. The idea behind it is that accumulating a rate gives the total change: if v(t) is velocity, then the integral of v from a to b is the net change in position. Differentiation and integration undo each other, much like squaring and square roots.
Example: to find the area under y = x² from 0 to 3, take F(x) = x³/3, so the area is F(3) − F(0) = 9 − 0 = 9. For Part 1, the derivative of the integral from 1 to x of t³ dt is x³, and with an upper limit of x² the chain rule gives 2x · (x²)³ = 2x⁷.
The theorem is the heart of calculus, powering physics, economics and engineering calculations. Tips: the constant C cancels in definite integrals, always subtract in the order upper minus lower, and be careful with functions that are not continuous on the interval, where the theorem may not apply.
Key ideas
- ∫ f(x) dx from a to b = F(b) − F(a)
- F′ = f
- Links slope to area
Quick check
Use cases: where Fundamental Theorem of Calculus helps
- Homework: review the key idea behind Fundamental Theorem of Calculus before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Fundamental Theorem of Calculus without notes.
- Visual learning: play with the interactive lab to see how Fundamental Theorem of Calculus changes when you adjust its inputs.
- Teaching a friend: explain Fundamental Theorem of Calculus in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Fundamental Theorem of Calculus?
Differentiation and integration are inverse operations.
What will I learn in Fundamental Theorem of Calculus?
∫ f(x) dx from a to b = F(b) − F(a). F′ = f. Links slope to area.
Is Fundamental Theorem of Calculus part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Fundamental Theorem of Calculus lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Fundamental Theorem of Calculus?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
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