 Midpoint Riemann Sum - Maple Programming Help

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Midpoint Riemann Sum

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

$\sum _{i=1}^{N}f\left(\frac{{x}_{i-1}}{2}+\frac{{x}_{i}}{2}\right)\left({x}_{i}-{x}_{i-1}\right)$

 where the chosen point of each subinterval $\left({x}_{i-1},{x}_{i}\right)$ of the partition is the midpoint $\frac{\left({x}_{i-1}+{x}_{i}\right)}{2}$.
 • 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

 > with(Student[Calculus1]):
 > RiemannSum(sin(x), x=0.0..5.0, method = midpoint);
 ${0.7238544375}$ (1)
 > RiemannSum(x*(x - 2)*(x - 3), x=0..5, method = midpoint, output = plot); > RiemannSum(tan(x) - 2*x, -1..1, method = midpoint, output = plot,            partition = 20, boxoptions=[filled=[color="Burgundy"]]); 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.

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