Course: Calculus I
Topic: Antiderivatives
Subtopic: Fundamental Theorems of Calculus


The Fundamental Theorem of Calculus (FTC) is key to making the connections between differentiation and integration. There are actually two forms of FTC. Undertanding FTC requires algebraic skills and visual observations. Supporting content is also included here, eg. properties of integrals.


By the end of this topic you should know and be prepared to be tested on:


Terms you should be able to define: integrable, integrability, net area, and ...

The First Fundamental Theorem of Calculus (FTC1)

If a function f is continuous on [a,b] and F is an antiderivative
of f on the interval [a,b], then `int_a^b f(x) dx = F(b) - F(a)`.

The Second Fundamental Theorem of Calculus (FTC2)

If f is continuous on an open interval I containing a,
then for every x in the interval, `d/dx [int_a^x f(x) dx] = f(x)`.

Note: I'm using the traditional titles of FTC1 and FTC2, but some authors reverse them or call them by different names. Several authors/websites call FTC2 "Fundamental Theorem of Calculus (part 1)" and FTC1 "Fundamental Theorem of Calculus (part 2)" including the Briggs/Cochran text.

Mini-Lectures and Examples

STUDY: Properties of Integrals, Net Area, Area Function, and FTC

Supplemental Resources (recommended)

Explore area as a function with The Area Function applet.

Two more applets for exploration: FTC 1 (theoretical part) and FTC 2 (practical part)

Useful video from 3Blue1Brown's The Essence of Calculus: Integration and the FTC

Supplemental Resources (optional)

Video: The Integral, Selwyn Hollis's Video Calculus

Video: The Fundamental Theorem of Calculus, Selwyn Hollis's Video Calculus

Lesson: Properties of the Definite Integral, Dale Hoffman's Contemporary Calculus

Lesson: The Fundamental Theorem of Calculus, Dale Hoffman's Contemporary Calculus

rev. 2020-11-27