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[parent] left hand rule (Theorem)

The left hand 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 \left( a+(j-1) \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+(j-1) \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 $ n$ 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 left endpoints of each of these intervals. Note that this is a special case of a left Riemann sum in which the $ x_j$'s are evenly spaced.



"left hand rule" is owned by Wkbj79.
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See Also: right hand rule, midpoint rule, Riemann sum, example of estimating a Riemann integral


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Cross-references: left Riemann sum, intervals, endpoints, function, partition, limit, rectangles, expressions, curve, area, measure, Riemann integral
There are 4 references to this entry.

This is version 13 of left hand rule, born on 2006-06-08, modified 2008-03-12.
Object id is 7974, canonical name is LeftHandRule.
Accessed 3581 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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