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Line 81: − ===Simpson's Rule=== − {{Main|Simpson's rule}} − '''Simpson's Rule''' is an extension of the Riemann integral. Instead of using shapes such as rectangles or trapezoids, Simpson's Rule allows for the use of parabolas or other higher order polynomials to approximate integrals.
fix headings
{{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.
===Simpson's Rule===
{{Main|Simpson's rule}}
'''Simpson's Rule''' is an extension of the Riemann integral. Instead of using shapes such as rectangles or trapezoids, Simpson's Rule allows for the use of parabolas or other higher order polynomials to approximate integrals.
==Double integrals==
==Double integrals==
:<math>\iint\limits_D f(x,y) dA = \int \int f(x,y) dx dy = \int \int f(x,y) dy dx</math>
:<math>\iint\limits_D f(x,y) dA = \int \int f(x,y) dx dy = \int \int f(x,y) dy dx</math>
==Lebesgue Integral==
==Lebesgue Integral==