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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=a=x0,x1,...,xN=b of the interval a,b, the midpoint Riemann sum is defined as:

i=1Nf12xi1+12xixixi1

  

where the chosen point of each subinterval xi1,xi of the partition is the midpoint 12xi1+xi.

• 

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

withStudent[Calculus1]:

RiemannSumsinx,x=0.0..5.0,method=midpoint

0.7238544375

(1)

RiemannSumxx2x3,x=0..5,method=midpoint,output=plot,boxoptions=filled=color=pink,transparency=0.5

RiemannSumtanx2x,1..1,method=midpoint,output=plot,partition=50

RiemannSumlnx,x=1..100,method=midpoint,output=animation

Other Riemann Sums

  

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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