# Fundamental theorem of calculus

  This article/section deals with mathematical concepts appropriate for late high school or early college.

## The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus, first proven by James Gregory, is the rather remarkable result that the two fundamental operations of calculus are just inverses of each other. Those two operations are performed on functions from the real numbers to the real numbers, and are most easily visualized when the functions are expressed in terms of graphs. The operations are:

• Differentiation—find the slope of a function's graph at a given point.
• Integration—find the area under a graph between two given limits.

## The Theorem

There are two parts to the Fundamental Theorem of Calculus[1]

### Part 1

The first can be written as: Let the function  be continuous function defined on a closed interval . Define  as:



It follows that:

The first part states that if a function  is the antiderivative of a function , then the derivative of  is . In other words, antiderivative and derivative are opposite functions.

### Part 2

If:



Then:



The second part begins with what we know from part 1.
It then states that the definite integral of the function  from  to  is equal to  evaluated at  minus  evaluated at .