The Fundamental Theorems of Calculus

Click For Summary
The Fundamental Theorems of Calculus establish the relationship between differentiation and integration, asserting that they are inverse processes. The first theorem states that if a function is continuous, its integral serves as an antiderivative. The second theorem indicates that if a function has a derivative equal to another function almost everywhere, they differ by a constant, provided the first function is Lipschitz continuous. The discussion highlights the confusion surrounding these concepts, particularly regarding the proofs and implications of the theorems. Understanding these theorems is crucial for grasping the foundations of calculus and its applications.
  • #31
briefly put, if f is only integrable, then every integral of f is also an antiderivative, but not every (continuous) antiderivative of f is an integral of f.
 

Similar threads

B Question about the fundamental theorem of calculus
  • · Replies 2 ·
Replies
2
Views
3K
B A proof of the fundamental theorem of calculus
  • · Replies 6 ·
Replies
6
Views
2K
I Questions about the fundamental thoerem of calculus
  • · Replies 10 ·
Replies
10
Views
2K
MHB Fundamental theorem of calculus and more....
  • · Replies 9 ·
Replies
9
Views
2K
B Some help understanding integrals and calculus in general
  • · Replies 42 ·
2
Replies
42
Views
5K
I Visual interpretation of Fundamental Theorem of Calculus
  • · Replies 5 ·
Replies
5
Views
2K
Fundamental Theorem of Calculus: Part One
  • · Replies 12 ·
Replies
12
Views
2K
I How Can You Integrate x-Squared Without the Fundamental Theorem of Calculus?
  • · Replies 22 ·
Replies
22
Views
4K
I How can you prove the integral without knowing the derivative?
  • · Replies 6 ·
Replies
6
Views
2K
I What would a calculus author have to say on ##\int r^2dm##?
  • · Replies 11 ·
Replies
11
Views
2K