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Line 1: − '''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:+ + + + + + + + + + + + + + + + + + + + + Line 11:
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Fundamental theorem of calculus (view source)
Revision as of 04:21, December 30, 2008
, 04:21, December 30, 2008added content and other major revisions
==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.
*[[Differentiation]] -- find the slope of a function's graph at a given point.
*[[integral|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 between x=a and x=b 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 Theorem==
There are two parts to the Fundamental Theorem of Calculus
===Part 1===
The first can be written as:
Let the function <math>f</math> be continuous function defined on a closed interval <math>[a,b]</math>. Define <math>F(x)</math> as:
<br />
<math>F(x) = \int_a^x f(t)dt</math>
<br />It follows that:
<br /><math>F'(x) = f(x)</math>
===Part 2===
If:
<br /><math>f(x) = F'(x)</math>
<br />Then:
<br /><math>\int_a^b f(x) = F(b) - F(a)</math>
==Indefinite Integrals==
{{main|Indefinite integral}}
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:
The Fundamental Theorem of Calculus says that the area under the graph of <math>x^2</math> between a and b is the difference in the values of <math>\frac{x^3}{3}+C</math> between a and b. Note that the constant of integration cancels out.
The Fundamental Theorem of Calculus says that the area under the graph of <math>x^2</math> between a and b is the difference in the values of <math>\frac{x^3}{3}+C</math> between a and b. Note that the constant of integration cancels out.
==Definite Integrals==
{{main|Integral}}
This kind of integral is called a ''definite integral'', written with the limits:
This kind of integral is called a ''definite integral'', written with the limits:
The above is a simplified "intuitive" treatment of calculus and of this theorem. The actual "rigorous" proof, "rigorous" definitions of derivative and integral, and statement of the conditions under which the theorem is true, are beyond the scope of this article.
The above is a simplified "intuitive" treatment of calculus and of this theorem. The actual "rigorous" proof, "rigorous" definitions of derivative and integral, and statement of the conditions under which the theorem is true, are beyond the scope of this article.
==See also==
*[[Calculus]]
*[[Indefinite integral]]
*[[Derivative]]
*[[Integral]]
[[Category:Calculus]]
[[Category:Calculus]]