right hand rule RightHandRule 2006-06-08 16:27:23 2008-03-12 07:58:14 Theorem Riemann integral \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} The \emph{right hand rule} for computing the Riemann integral $\displaystyle \int\limits_a^b f(x) \, dx$ is \[ \int\limits_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{j=1}^n f \left( a+j \left( \frac{b-a}{n} \right) \right) \left( \frac{b-a}{n} \right). \] If the Riemann integral is considered as a measure of area under a curve, then the expressions $\displaystyle f \left( a+j \left( \frac{b-a}{n} \right) \right)$ \PMlinkescapetext{represent} the \PMlinkescapetext{heights} of the rectangles, and $\displaystyle \frac{b-a}{n}$ is the common \PMlinkescapetext{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 \PMlinkescapetext{right} endpoints of each of these intervals. Note that this is a special case of a \PMlinkname{right Riemann sum}{RightRiemannSum} in which the $x_j$'s are evenly spaced.