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Added indefinite integrals; double integrals
[[Image:Definiteintegralnv7.gif|thumb|A definite integral.]]
[[Image:Definiteintegralnv7.gif|thumb|A definite integral.]]
[[Indefinite integral]]s use the notation
:<math>\int f(x) dx </math>
==Integration==
==Integration==
Integration is the inverse function of the [[derivative]] and the two notions are related by the [[Fundamental Theorem of Calculus]].
Integration is the inverse function of [[differentiation]]; the two notions are related by the [[Fundamental Theorem of Calculus]]. Thus, if <math>\int f(x) dx = g(x)</math>, then <math>\frac{d}{dx}g(x) = f(x)</math>.
Note that, because the derivative of any real number is 0, <math>\frac{d}{dx}x^2+2 = \frac{d}{dx}x^2+3 = \frac{d}{dx}x^2 = 2x</math>. Therefore, <math>\int 2x dx</math> does not simply equal x<sup>2</sup>, but rather x<sup>2</sup> + C for any constant real number C. The above-mentioned functions are the ''family of antiderivatives'' for 2x.
The concept of integration can be extended to functions in more than one real variable, as well as functions defined over the [[complex numbers]].
The concept of integration can be extended to functions in more than one real variable, as well as functions defined over the [[complex numbers]].
==Antiderivative vs Integration==
==Antiderivative vs Integration==
There are important differences between the anti-derivative and integration. An anti-derivative of a function <math>f(x)</math> is a function <math>g(x)</math> such that,
There are important differences between the anti-derivative and integration. An anti-derivative of a function <math>f(x)</math> is a function <math>F(x)</math> such that,
:<math>\frac{d}{dx}F(x)=f(x)</math>
:<math>\frac{d}{dx}F(x)=f(x)</math>
{{main|Riemann Integral}}
{{main|Riemann Integral}}
As a geometric interpretation of the integral of the [[area]] under a curve, the Riemann integral consists of dividing the area under the curve of the function into slices. The [[domain]] of the function is partioned into N segments of width <math>\frac{b-a}{N}</math>. The height of the segment is dependent on which side of the rectangle is taken. The lower sum takes the lower side of the rectangle, the upper sum the higher side of the rectangle. In the [[limit]] of <math>N\rightarrow\infty</math> these two [[series (mathematics)|series]] become the integral. If they approach the same value then the integral exists, otherwise it is undefined.
As a geometric interpretation of the integral of the [[area]] under a curve, the Riemann integral consists of dividing the area under the curve of the function into slices. The [[domain]] of the function is partioned into N segments of width <math>\frac{b-a}{N}</math>. The height of the segment is dependent on which side of the rectangle is taken. The lower sum takes the lower side of the rectangle, the upper sum the higher side of the rectangle. In the [[limit]] of <math>N\rightarrow\infty</math> these two [[series (mathematics)|series]] become the integral. If they approach the same value then the integral exists, otherwise it is undefined.
==Double integrals==
Double integrals are integrals extended to two dimensions. Just like a definite integral gives the area under a 2-dimensional function, a double integral gives the volume under a three-dimensional function. A double integral is not integrated over a single variable x, but rather over an area, usually written A. It is written
:<math>\iint\limits_D f(x,y) dA</math>
[[Fubini's Theorem]] states that
:<math>\iint\limits_D f(x,y) dA = \int \int f(x,y) dx dy = \int \int f(x,y) dy dx</math>
===Simpson's Rule===
===Simpson's Rule===