Riemann Sums

Numerical estimates to definite integrals can be obtained via various types of summations. This page provides

A visualization illustrating why summation should work and why some techniques are more precise than others.
A numerical tool for actually computing these sums.

Visualization

This demo allows you to choose a function, partition size, and sum type (from Left, Right, Midpoint, or Trapezoidal) to compute and visualize the corresponding numerical integration approximation.

viewof example = Inputs.select(
  [
    {
      label: "linear",
      a: 0,
      b: 4,
      f: x => 1 + x/2,
      f_tex: String.raw`\left(1+x/2\right)`,
      actual: 8
    },
    {
      label: "quadratic",
      a: 0,
      b: 2,
      f: x => 2 - x**2/2,
      f_tex: String.raw`\left(2 - \frac{1}{2}x^2\right)`,
      actual: 2.666666666
    },
    {
      label: "cubic",
      a: -3,
      b: 4,
      f_tex: "(x^3 - 5x - 2)",
      f: x => x**3 - 5*x - 2,
      actual: 12.25
    },
    {
      label: "exponential",
      a: 0,
      b: 3,
      f_tex: "e^{-x}",
      f: x => Math.exp(-x),
      actual: 0.950212931632137
    },
    {
      label: "normal",
      a: -2,
      b: 2,
      f_tex: String.raw`\frac{1}{\sqrt{2\pi}} e^{-x^2/2}`,
      f: x => Math.exp(-x*x/2)/Math.sqrt(2*Math.PI),
      actual: 0.9544997361036
    },
    {
      label: "crazy sine",
      a: 0,
      b: 3,
      f_tex: String.raw`\sin(x^3)`,
      f: x => Math.sin(x**3),
      actual: 0.45642404618707
    }
  ],
  {format: o => o.label, label: "Example:"});

viewof type = Inputs.radio(
  ["Left", "Right", "Midpoint", "Trapezoidal"],
  {label: "Sum type:", value: "Left"}
)

viewof N = Inputs.range([1,100], {label: tex`N:`, value: 10, step:1})

tex.block`${
  String.raw`\int_{${example.a}}^{${example.b}} ${example.f_tex} \, dx 
  \approx ${d3.format('0.5f')(estimate)} \: \: \: \: \: \: \text{error} \approx ${
  d3.format("0.5f")(Math.abs(example.actual - estimate))}`
}`

riemann_plot = plot(example);

update = riemann_plot.update_rects(type, N)

estimate = compute_sum(example, N, type);

Computation

Here’s some live SageMath code that allows you to actually compute these sums.

\[\int_a^b f(x) \, dx \approx \sum_{i=1}^n f(x_{i-1}) \, \Delta x\]

\[\int_a^b f(x) \, dx \approx \sum_{i=1}^n f(x_i) \, \Delta x\]

\[\int_a^b f(x) \, dx \approx \sum_{i=1}^n f\left(\frac{x_{i-1} + x_i}{2}\right) \, \Delta x\]

\[\int_a^b f(x) \, dx \approx \sum_{i=1}^n \frac{f(x_{i-1}) + f(x_i)}{2} \, \Delta x\]

Note how similar the computer code for the sum is to the mathematical notation. That is

sum(f(x(i))*dx, i,1,n) looks like
\(\sum_{i=1}^n f(x_i) \, \Delta x\)

import {plot} from './components/plot.js'
import {compute_sum} from './components/estimate_integral.js'