PlanetMath: midpoint rule

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midpoint rule

(Theorem)

The midpoint rule for computing the Riemann integral $\displaystyle \int\limits_a^b f(x) \, dx$ is

$\displaystyle \int\limits_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{j=1}^n f \... ...{2} \right) \left( \frac{b-a}{n} \right) \right) \left( \frac{b-a}{n} \right).$

If the Riemann integral is considered as a measure of area under a curve, then the expressions $\displaystyle f \left( a + \left( j-\frac{1}{2} \right) \left( \frac{b-a}{n} \right) \right)$ represent the heights of the rectangles, and $\displaystyle \frac{b-a}{n}$ is the common width of the rectangles.

The Riemann integral can be approximated by using a definite value for rather than taking a limit. In this case, the partition is $\displaystyle \left\{ \left[ a, a+\frac{b-a}{n} \right) , \dots , \left[ a+\frac{(b-a)(n-1)}{n}, b \right] \right\}$ , and the function is evaluated at the midpoints of each of these intervals. Note that this is a special case of a Riemann sum in which the 's are evenly spaced and the 's chosen are the midpoints.

If is Riemann integrable on such that $\vert f''(x)\vert \le M$ for every $x \in [a,b]$ , then

$\displaystyle \left\vert \int\limits_a^b f(x) \, dx - \sum_{j=1}^n f \left( a +... ...) \right) \left( \frac{b-a}{n} \right) \right\vert \le \frac{M(b-a)^3}{24n^2}.$

"midpoint rule" is owned by Wkbj79.

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Cross-references: Riemann integrable, Riemann sum, intervals, midpoints, function, partition, limit, rectangles, expressions, curve, area, measure, Riemann integral
There are 3 references to this entry.

This is version 13 of midpoint rule, born on 2006-06-08, modified 2008-03-12.
Object id is 7976, canonical name is MidpointRule.
Accessed 7324 times total.

Classification:

AMS MSC:	26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)
	41-01 (Approximations and expansions :: Instructional exposition )
	28-00 (Measure and integration :: General reference works )

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