Lower Riemann Sum - Maple Programming Help

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Lower Riemann Sum

 Calling Sequence RiemannSum(f(x), x = a..b, method = lower, opts) RiemannSum(f(x), a..b, method = lower, opts) RiemannSum(Int(f(x), x = a..b), method = lower, 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 = lower, opts) command calculates the lower 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 lower Riemann sum is defined as:

$\sum _{i=1}^{N}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}f\left({x}_{i}^{*}\right)\left({x}_{i}-{x}_{i-1}\right)$

 where the chosen point of each subinterval $\left({x}_{i-1},{x}_{i}\right)$ of the partition is a point ${x}_{i}^{*}$ such that $f\left(x\right)\ge f\left({x}_{i}^{*}\right)$ for all $x$ in $\left({x}_{i-1},{x}_{i}\right)$.
 • 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

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{Calculus1}]\right):$
 > $\mathrm{RiemannSum}\left(\mathrm{sin}\left(x\right),x=0.0..5.0,\mathrm{method}=\mathrm{lower}\right)$
 ${-}{0.0576648694}$ (1)
 > $\mathrm{RiemannSum}\left(x\left(x-2\right)\left(x-3\right),x=0..5,\mathrm{method}=\mathrm{lower},\mathrm{output}=\mathrm{plot},\mathrm{boxoptions}=\left[\mathrm{filled}=\left[\mathrm{color}=\mathrm{pink},\mathrm{transparency}=0.5\right]\right]\right)$
 > $\mathrm{RiemannSum}\left(\mathrm{tan}\left(x\right)-2x,x=-1..1,\mathrm{method}=\mathrm{lower},\mathrm{output}=\mathrm{plot},\mathrm{partition}=50\right)$

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

 > $\mathrm{RiemannSum}\left(\mathrm{ln}\left(x\right),1..100,\mathrm{method}=\mathrm{lower},\mathrm{output}=\mathrm{animation}\right)$
 Other Riemann Sums