JavaScript/Maths/MidpointIntegration.js

/**
*
* @title Midpoint rule for definite integral evaluation
* @author [ggkogkou](https://github.com/ggkogkou)
* @brief Calculate definite integrals with midpoint method
*
* @details The idea is to split the interval in a number N of intervals and use as interpolation points the xi
* for which it applies that xi = x0 + i*h, where h is a step defined as h = (b-a)/N where a and b are the
* first and last points of the interval of the integration [a, b].
*
* We create a table of the xi and their corresponding f(xi) values and we evaluate the integral by the formula:
* I = h * {f(x0+h/2) + f(x1+h/2) + ... + f(xN-1+h/2)}
*
* N must be > 0 and a<b. By increasing N, we also increase precision
*
* [More info link](https://tutorial.math.lamar.edu/classes/calcii/approximatingdefintegrals.aspx)
*
*/

function integralEvaluation (N, a, b, func) {
  // Check if all restrictions are satisfied for the given N, a, b
  if (!Number.isInteger(N) || Number.isNaN(a) || Number.isNaN(b)) { throw new TypeError('Expected integer N and finite a, b') }
  if (N <= 0) { throw Error('N has to be >= 2') } // check if N > 0
  if (a > b) { throw Error('a must be less or equal than b') } // Check if a < b
  if (a === b) return 0 // If a === b integral is zero

  // Calculate the step h
  const h = (b - a) / N

  // Find interpolation points
  let xi = a // initialize xi = x0
  const pointsArray = []

  // Find the sum {f(x0+h/2) + f(x1+h/2) + ... + f(xN-1+h/2)}
  let temp
  for (let i = 0; i < N; i++) {
    temp = func(xi + h / 2)
    pointsArray.push(temp)
    xi += h
  }

  // Calculate the integral
  let result = h
  temp = 0
  for (let i = 0; i < pointsArray.length; i++) temp += pointsArray[i]

  result *= temp

  if (Number.isNaN(result)) { throw Error('Result is NaN. The input interval does not belong to the functions domain') }

  return result
}

export { integralEvaluation }