3 The Fundamental Theorem of Calculus

Note 5.2 - The Fundamental Theorem of Calculus

1 Introduction

The motivation to study the fundamental theorem is probably to compute inte- gration. But the theorem itself is probably the most important and fundamental fact to know in calculus.

2 The Area Functions

Given an continuous function f (x) and some a ∈ R, we know thatRx

a f (x) dx is always defined for x ≥ a. For x < a, we define

Z x a

f (x) dx := − Z a

x

f (x) dx.

This gives us a well-defined area function F (x) =

Z x a

f (x) dx

on R. We have the following properties that follows directly from definition of F :

If we can explicitly find this function F , the tedious computations in Note 5.1 become much simpler:

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So how to find this F ?

3 The Fundamental Theorem of Calculus

We now come to the most important theorem of the entire course: finding the relationship between F and f described above.

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Theorem 3.1 (Fundamental Theorem of Calculus FTC). Given a continuous function f and

F (x) :=

Z x s

f (t) dt for some s ∈ R, then

• F is differentiable and F⁰= f

•

Z b a

f (x) dx = F (b) − F (a).

So this gives us much more clues on how to find F . We will derive (more precisely, guess) some formula in the next note.

Here is a more general form of FTC:

Theorem 3.2 (Generalized Fundamental Theorem of Calculus). Given a con- tinuous function f and differentiable functions g, h, let

F (x) = Z g(x)

h(x)

f (t) dt.

Then,

F⁰(x) = f (h(x))h⁰(x) − f (g(x))g⁰(x).

It is an easy consequence of the chain rule:

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4 Examples

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