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:

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.

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"]]);

exact := int(ln(x), x=1..100);

$\mathrm{exact}≔-99+200\ln \left(2\right)+200\ln \left(5\right)$

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