Right Riemann Sum - Maple Help

Right Riemann Sum

 Calling Sequence RiemannSum(f(x), x = a..b, method = right, opts) RiemannSum(f(x), a..b, method = right, opts) RiemannSum(Int(f(x), x = a..b), method = right, opts)

Parameters

 f(x) - algebraic expression in variable 'x' x - name; specify the independent variable a, b - algebraic expressions; specify the interval opts - equation(s) of the form option=value where option is one of boxoptions, functionoptions, iterations, method, outline, output, partition, pointoptions, refinement, showarea, showfunction, showpoints, subpartition, view, or Student plot options; specify output options

Description

 • The RiemannSum(f(x), x = a..b, method = right, opts) command calculates the right Riemann sum of f(x) from a to b. The first two arguments (function expression and range) can be replaced by a definite integral.
 • If the independent variable can be uniquely determined from the expression, the parameter x need not be included in the calling sequence.
 • Given a partition $P=\left(a={x}_{0},{x}_{1},...,{x}_{N}=b\right)$ of the interval $\left(a,b\right)$, the right Riemann sum is defined as:

$\sum _{i=1}^{N}f\left({x}_{i}\right)\left({x}_{i}-{x}_{i-1}\right)$

 where the chosen point of each subinterval $\left({x}_{i-1},{x}_{i}\right)$ of the partition is the right-hand point ${x}_{i}$.
 • By default, the interval is divided into $10$ equal-sized subintervals.
 • For the options opts, see the RiemannSum help page.
 • This integration method can be applied interactively, through the ApproximateInt Tutor.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right):$
 > $\mathrm{RiemannSum}\left(\mathrm{sin}\left(x\right),x=0...5.0,\mathrm{method}=\mathrm{right}\right)$
 ${0.4616204858}$ (1)
 > $\mathrm{RiemannSum}\left(x\left(x-2\right)\left(x-3\right),x=0..5,\mathrm{method}=\mathrm{right},\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{RiemannSum}\left(\mathrm{tan}\left(x\right)-2x,x=-1..1,\mathrm{method}=\mathrm{right},\mathrm{output}=\mathrm{plot},\mathrm{partition}=20,\mathrm{boxoptions}=\left[\mathrm{filled}=\left[\mathrm{color}="Burgundy"\right]\right]\right)$

To play the following animation in this help page, right-click (on Macintosh, Control-click) the plot to display the context menu.  Select Animation > Play.

 > $\mathrm{exact}≔\mathrm{int}\left(\mathrm{ln}\left(x\right),x=1..100\right)$
 ${\mathrm{exact}}{≔}{-}{99}{+}{200}{}{\mathrm{ln}}{}\left({2}\right){+}{200}{}{\mathrm{ln}}{}\left({5}\right)$ (2)
 > $\mathrm{evalf}\left(\mathrm{exact}\right)$
 ${361.5170185}$ (3)
 > $\mathrm{RiemannSum}\left(\mathrm{ln}\left(x\right),1..100,\mathrm{method}=\mathrm{right},\mathrm{outline}=\mathrm{true},\mathrm{output}=\mathrm{animation}\right)$
 Other Riemann Sums