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| {{merge|Riemann Integral|date=February 2012}}
| | Specialist Managers Harry from Bedford, likes to spend time walking and hiking, [http://www.biclines.com/ActivityFeed/MyProfile/tabid/60/userId/28201/language/en-US/Default.aspx private property developers in singapore] developers in singapore and crocheting. These days took some time to travel to Macquarie Island. |
| {{Unreferenced|date=December 2009}}
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| In [[mathematics]], specifically in [[integral calculus]], the '''rectangle method''' (also called the ''midpoint'' or ''mid-ordinate rule'') computes an [[approximation]] to a [[definite integral]], made by finding the [[area]] of a collection of [[rectangle]]s whose heights are determined by the values of the function.
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| Specifically, the interval <math>(a,b)</math> over which the function is to be integrated is divided into <math>N</math> equal subintervals of length <math>h = (b-a)/N</math>. The rectangles are then drawn so that either their left or right corners, or the middle of their top line lies on the [[graph of a function|graph]] of the function, with bases running along the <math>x</math>-axis. The approximation to the integral is then calculated by adding up the areas (base multiplied by height) of the <math>N</math> rectangles, giving the formula:
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| :<math>\int_a^b f(x)\,dx \approx h \sum_{n=0}^{N-1} f(x_{n})</math>
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| where <math>h = (b - a) / N</math> and <math>x_{n} = a + nh </math>.
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| The formula for <math>x_{n}</math> above gives <math>x_{n}</math> for the Top-left corner approximation.
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| As ''N'' gets larger, this approximation gets more accurate. In fact, this computation is the spirit of the definition of the [[Riemann integral]] and the [[Limit of a sequence|limit]] of this approximation as <math>n \to \infty</math> is defined and equal to the integral of <math>f</math> on <math>(a,b)</math> if this Riemann integral is defined. Note that this is true regardless of which <math>i'</math> is used, however the midpoint approximation tends to be more accurate for finite <math>n</math>.
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| {{Gallery
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| |title=The different rectangle approximations
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| ||Top-left corner approximation
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| |Image:midRiemann.png|Midpoint approximation
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| ||Top-right corner approximation
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| }}
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| ==Error==
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| For a function <math>f</math> which is twice differentiable, the approximation error in each section <math>(a,a+\Delta)</math> of the midpoint rule decays as the cube of the width of the rectangle. (For a derivation based on a Taylor approximation, see [[Midpoint method]])
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| :<math>E_i \le \frac{\Delta^3}{24}\,f''(\xi) </math>
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| for some <math>\xi</math> in <math>(a, a+\Delta)</math>. Summing this, the approximation error for <math>n</math> intervals with width <math>\Delta</math> is less than or equal to
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| :<math>n=1,2,3,...</math>
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| where <math>n+1</math> is the number of nodes
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| :<math>E \le \frac{n\Delta^3}{24}f''(\xi) </math>
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| in terms of the total interval, we know that <math>n\Delta=b-a</math> so we can rewrite the expression:
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| :<math>E \le \frac{(b-a)\Delta^2}{24}f''(\xi) </math>
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| for some <math>\xi</math> in <math>(a,b)</math>.
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| ==See also==
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| * [[Midpoint method]] for solving [[ordinary differential equation]]s
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| * [[Trapezoidal rule]]
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| * [[Simpson's rule]]
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| {{DEFAULTSORT:Rectangle Method}}
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| [[Category:Integral calculus]]
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| [[Category:Numerical integration (quadrature)]]
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Specialist Managers Harry from Bedford, likes to spend time walking and hiking, private property developers in singapore developers in singapore and crocheting. These days took some time to travel to Macquarie Island.