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'''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:

'''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.

*[[Differentiation]] -- find the slope of a function's graph at a given point.

*Integration -- find the area under a graph between two given limits.

*[[integral|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 is just g(b)-g(a).

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 antiderivative of a function is often called the ''indefinite integral''.  (Indefinite because the limits a and b haven't been specified.)  So, for example, the derivative of <math>\frac{x^3}{3}+7</math> is <math>x^2</math>.  From this it follows that the antiderivative of  

The antiderivative of a function is often called the ''indefinite integral''.  (Indefinite because the limits a and b haven't been specified.)  So, for example, the derivative of <math>\frac{x^3}{3}+7</math> is <math>x^2</math>.  From this it follows that the antiderivative of  

<math>x^2</math> could be <math>\frac{x^3}{3}+7</math>.  But note that the "7" in that formula was a red herring.  Adding any constant to a function doesn't change its derivative, so the antiderivative of <math>x^2</math> could have any constant added to it.  This arbitrary constant is usually written '''C''' and is called the "constant of integration.  The indefinite integral could be written:

<math>x^2</math> could be <math>\frac{x^3}{3}+7</math>.  But note that the "7" in that formula was a red herring.  Adding any constant to a function doesn't change its derivative, so the antiderivative of <math>x^2</math> could have any constant added to it.  This arbitrary constant is usually written '''C''' and is called the "constant of integration".  The indefinite integral could be written:

:<math>\int x^2\ \mathrm{d}x = \frac{x^3}{3} + C\,</math>,

:<math>\int x^2\ \mathrm{d}x = \frac{x^3}{3} + C\,</math>,