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).

As a mathematical statement the fundametal theorem of calculus read,

<math>\int^{b}_{a}\frac{d}{dx}F(x)dx=F(b)-F(a)</math>

For many years calculus focused on finding the anti-derivative of a function in order to integrate it. It is interesting, however, that for some functions, such as <math>e^{x^{2}}</math>, the anti-derivative cannot be expressed in terms of elementary functions, such as sines, logs, square roots, etc.