Fundamental theorem of calculus

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 Fundamental Theorem of Calculus says that the two operations are inverses -- to find the area under the graph of f(x) between a and b, find the function g(x) whose derivative is f(x) (that is, find the antiderivative of f). The area under the graph of f between x=a and x=b is just g(b)-g(a).

The Theorem

There are two parts to the Fundamental Theorem of Calculus

Part 1

The first can be written as: Let the function $f$ be continuous function defined on a closed interval $[a,b]$. Define $F(x)$ as:
$F(x)=\int _{a}^{x}f(t)dt$
It follows that:
$F'(x)=f(x)$
The first part states that if a function $F$ is the antiderivative of a function $f$, then the derivative of $F$ is $f$. In other words, antiderivative and derivative are opposite functions.

Part 2

If:
$f(x)=F'(x)$
Then:
$\int _{a}^{b}f(x)=F(b)-F(a)$
The second part begins with what we know from part 1.
It then states that the definite integral of the function $f$ from $a$ to $b$ is equal to $F$ evaluated at $b$ minus $F$ evaluated at $a$.

See also

• Calculus
• Indefinite integral
• Derivative
• Integral