Midpoint Riemann Sum
Calling Sequence
Parameters
Description
Examples
Other Riemann Sums
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)
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
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=a=x0,x1,...,xN=b of the interval a,b, the midpoint Riemann sum is defined as:
∑i=1N⁡f⁡xi−12+xi2⁢xi−xi−1
where the chosen point of each subinterval xi−1,xi of the partition is the midpoint xi−1+xi2.
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.
with(Student[Calculus1]):
RiemannSum(sin(x), x=0.0..5.0, method = midpoint);
0.7238544375
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);
exact≔−99+200⁢ln⁡2+200⁢ln⁡5
evalf(exact);
361.5170185
RiemannSum(ln(x), x=1..100, method = midpoint, outline = true, output = animation);
Left Riemann Sum
Lower Riemann Sum
Right Riemann Sum
Upper Riemann Sum
See Also
int
plot/options
Student
Student plot options
Student[Calculus1]
Student[Calculus1][ApproximateInt]
Student[Calculus1][ApproximateIntTutor]
Student[Calculus1][RiemannSum]
Student[Calculus1][VisualizationOverview]
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