Expert: Paul Klarreich - 3/11/2007

Question
What is the Fundamental Theorem of Calculus? Can you describe it first in "everyday language" and then give a mathematical description?  Can you do the same for the Second Fundamental Theorem of Calculus?

Answer
Questioner:   Tom
Category:  Calculus
 
Subject:  Fundamental Theorem of Calculus
Question:  What is the Fundamental Theorem of Calculus? Can you describe it first in "everyday language" and then give a mathematical description?  Can you do the same for the Second Fundamental Theorem of Calculus?
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Hi, Tom,

The FTC says, basically, that the Definite Integral and the Derivative are inverse operations.  For the mathematical description, see Chapter 4 of any university-level calculus text, available for a reasonable price second-hand at lots of good bookstores.

As to a Second FTC, I think that's a variant that just says that the derivative of a function defined as an integral, like this:

{ x
|   f(t) dt
} a  

is the integrand applied to x, that is: f(x)

Both of these are really saying the same thing -- differentiation and integration are inverses of each other.