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Line 17: − <h3>1. Additive Distribution</h3> − If an integral contains two functions added together, it may be rewritten as two separate integrals, each containing one function. − <h3>2. Constant Multiplicative Distribution</h3> − If an integral contains a function multiplied by a constant, it may be rewritten as the constant times the integral of the function. − <h3>3. Area</h3> − Any integral that contains the same upper and lower bounds are equal to zero. − <h3>4. Separation</h3> − An integral may be rewritten as the sum of two integrals with adjacent bounds. − <h3>5. Additive Inversion</h3> − Any integral is equal to the additive inverse of the same integral with reversed upper and lower bounds. Line 49:
Line 39: − {{main|Riemann Integral}}+ − The Riemann integral is a geometric interpretation [[area]] under a curve (also called the [[integral]]), the Riemann integral consists of dividing the area under the curve of the function into smaller slices. These do not necessarily have to be rectangles (for example, [[trapezoids]] or even [[parabolas]]), but in most cases − −
improper formatting, bad grammar + subsequent edit seemed widly incorrect
Intergation has the following properties<ref>[http://www.sosmath.com/calculus/integ/integ02/integ02.html Properties of Intergrals]</ref>
Intergation has the following properties<ref>[http://www.sosmath.com/calculus/integ/integ02/integ02.html Properties of Intergrals]</ref>
*<math>\int_{a}^{b}(f(x)+g(x))dx=\int_{a}^{b}f(x)dx+\int_{a}^{b}g(x)dx</math>
*<math>\int_{a}^{b}(f(x)+g(x))dx=\int_{a}^{b}f(x)dx+\int_{a}^{b}g(x)dx</math>
*<math>\int_{a}^{b}cf(x)dx=c\int_{a}^{b}f(x)dx</math>
*<math>\int_{a}^{b}cf(x)dx=c\int_{a}^{b}f(x)dx</math>
*<math>\int_{a}^{a}f(x)dx=0</math>
*<math>\int_{a}^{a}f(x)dx=0</math>
*<math>\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx</math>, <math>c\in(a,b)</math>
*<math>\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx</math>, <math>c\in(a,b)</math>
*<math>\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx</math>
*<math>\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx</math>
==Riemann integral==
==Riemann integral==
As a geometric interpretation of the integral of the [[area]] of a curve, the Riemann integral consists of dividing the area under the curve of the function into rectangles. 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.
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.
==Lebesgue Integral==
==Lebesgue Integral==