RiemannSumSampler.sagews

The purpose of this worksheet is to help illustrate the concept of Riemann sum.

It allows you to compare, both numerically and graphically, $$\int_a^bf(x)dx\qquad \mbox{and} \qquad \sum_{i=1}^nf(x_i^*)(x_i-x_{i-1})=\left(\frac{b-a}{n}\right)\sum_{i=1}^nf(x_i^*)$$ namely the definite integral of any function $f$ over any interval $[a,b]$and its left $\left(\mbox{i.e.}\;x_i^*=x_{i-1}\right)$ / right $\left(\mbox{i.e.}\;x_i^*=x_i\right)$ / middle $\left(\mbox{i.e.}\;x_i^*=(x_{i-1}+x_i)/2\right)$ Riemann sum with respect to the partition $$x_i=a+i\left(\frac{b-a}{n}\right)\qquad i=0, 1, \ldots, n$$ of $[a,b]$ into $n$ subintervals of equal width.

First draft

In the cell below:

• Make your own choices for $f(x)$, $a$, $b$ and $n$.

• Set $t=0$ if you want a left Riemann sum, $t=1$ for a right one, and $t=0.5$ for a middle one.

• Evaluate and see the output.

f(x) = x^3-5*x+2 ### The function.
a = -1 ### The lower bound.
b = 2 ### The upper bound.
n = 18 ### The number of rectangles.
t = 1 ### 0 for a left Riemann sum, 1 for a right one, 0.5 for a middle one.
###############################################################################################
##########################Do not edit below this line.#########################################
###############################################################################################
I = integral(f(x), x, a, b).n()
delta = (b-a)/n; tdelta = t*delta; xk = a; L = []; S = 0
for k in range(n):
    L = L + [(xk, 0)]
    y = f(xk+tdelta)
    S = S + y
    L = L + [(xk,y)]
    xk = xk + delta
    L = L + [(xk, y)]
S = delta*S.n()
pretty_print('Integral = %s'%I)
pretty_print('Riemann Sum = %s'%S)
L = L + [(xk,0)]
G = plot(f(x), (x, a-(b-a)/20, b+(b-a)/20), color = 'red', thickness = 1)
G = G + plot(f(x), (x, a, b), color = 'red', thickness = 1, fill = true, fillcolor = 'grey')
G = G + polygon(L, edgecolor = 'black', rgbcolor = (t,t^2,1-t))
G.show(aspect_ratio = 'automatic')

Integral = 2.25000000000000

Riemann Sum = 1.77083333333333

Using the interactive capabilities of SageMath

Evaluating the following cell will do the same thing as above, with some of the user-friendly options provided by @interact. See http://wiki.sagemath.org/interact/ for more examples.

@interact
def _(f=input_box(default=x^3-5*x+2), a=input_box(default=-1), b=input_box(default=2), n=input_box(default=15), Method=['Left', 'Right', 'Middle']):
    if Method=='Left':
        t=0
    elif Method=='Right':
        t=1
    else:
        t=0.5
    f(x)=f
    I = integral(f(x), x, a, b).n()
    delta = (b-a)/n; tdelta = t*delta; xk = a; L = []; S = 0
    for k in range(n):
        L = L + [(xk, 0)]
        y = f(xk+tdelta)
        S = S + y
        L = L + [(xk,y)]
        xk = xk + delta
        L = L + [(xk, y)]
    S = delta*S.n()
    pretty_print('Integral = %s'%I)
    pretty_print('Riemann Sum = %s'%S)
    L = L + [(xk,0)]
    G = plot(f(x), (x, a-(b-a)/20, b+(b-a)/20), color = 'red', thickness = 1)
    G = G + plot(f(x), (x, a, b), color = 'red', thickness = 1, fill = true, fillcolor = 'grey')
    G = G + polygon(L, edgecolor = 'black', rgbcolor = (t,t^2,1-t))
    G.show(aspect_ratio = 'automatic')

Interact: please open in CoCalc

Sharing our interact with the help of SageMathCell

Entering the above code in SageMathCell allows us to share this interact easily. Here it is.