Difference between revisions of "Riemann Integral"

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(New page: ==Riemann Integral== As a geometric interpretation of the area under a curve (also called the integral), the Riemann integral consists of dividing the area under the curve of the f...)
 
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==Riemann Integral==
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As a geometric interpretation of the [[area]] under a curve (also called the [[integral]]), the Riemann integral consists of dividing the area under the curve of the function into fixed width slices. These do not necessarily have to be rectangles; in many cases using [[trapezoids]] or even [[parabolas]] for a Riemann integral is more useful.
 
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The [[domain]] of the function is partitioned into N segments of width. These segments are called ''subintervals'' of the partition.
 
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The width of each subinterval can be represented by
 
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<br /><math>\frac{b-a}{N}</math>
 
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To find the area under the curve, the areas of all subintervals are added together. With each shape of subintervals, there are different equations used to evaluate the area. If the subintervals are rectangles, we can use the properties of rectangles to find the area.  The area of a rectangle is height times width. To get the height of the rectangles, we evaluate the function at the subinterval.
 
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The equation for the area of each rectangle is
 
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<br /><math>A = f(x_i)\left ( \frac{b-a}{N} \right )</math>
 
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For simplification, we can say that
 
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*<math>\Delta x = \frac{b-a}{N}</math>
 
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*<math>A = f\left (x \right ) \Delta x</math>
 
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There are two methods of evaluating the area of each rectangle.  The function could be evaluated at the left '''or''' right side of each rectangle.
 
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Evaluated from the left side:
 
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<br /><math>\sum_{i=0}^N f\left (a+i\Delta x \right ) \Delta x</math>
 
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Evaluated from the right side:
 
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<br /><math>\sum_{i=1}^N f\left (a+i\Delta x \right ) \Delta x</math>
 
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If we increase the number of partitions, '''N''' the summation is a closer approximation of the area under the curve.  This can be written as
 
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<br />
 
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<math>\lim_{n \to \infty}\sum_{i=1}^N f\left (a+i\Delta x \right ) \Delta x = \int\limits_a^b f(x)dx</math>
 

Revision as of 02:44, January 17, 2010

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