Midpoint Rule (Rectangle Method) Algorithm is used for approximating definite integral.

This method works by approximating the area under the function (in given interval) as rectangle(s).

Midpoint Rule/Rectangle Method
Midpoint Rule/Rectangle Method With N = 10
Midpoint Rule/Rectangle Method With N = 20
Midpoint Rule/Rectangle Method With N = 20

As can be seen, the bigger the n (more intervals) is, the better the integral is approximated.


Midpoint Rule (Rectangle Method) Formula
Midpoint Rule (Rectangle Method) Formula
Midpoint Rule (Rectangle Method) M_i Formula
Midpoint Rule (Rectangle Method) M_i Formula


  • a is the beggining of the interval, b is the end of the interval,
  • n is the number of steps (number of rectangles to use),
  • f(x) is the function to integrate,
  • m_i is the base of i-th rectangle.


Let’s approximate this definite integral with n = 5.

Definite Integral 0-2 x*x dx.

Midpoint Formula (Rectangle Method) Example

Midpoint Formula (Rectangle Method) Example 2

Midpoint Formula (Rectangle Method) Example 3

As can be seen, for n = 5, we calculated that the definite integral is equal to 66/25 = 2.64, which is really close to the actual value: 8/3 = 2.(6).

Midpoint Rule (Rectangle Method) N Comparison

566/25 = 2.642/75 = 0.02(6)
10133/50 = 2.661/150 = 0.00(6)
251666/625 = 2.66562/1875 = 0.0010(6)


Function f, which is continous function, interval [a,b] and number of iterations to perform - n.
Defined Integral approximation of given function in given interval.
1. Calculate step = (b - a) / n. It will be used as the step size of each iteration.
2. Assign 0 to variable integral.
3. Initialize variable i to 1.
4. Add step * f(a + (i - 1) * step) to variable integral.
5. Increase i by one.
6. Check if i is smaller than or equal to n. If it is go to step 4.
7. End algorithm and return integral.

Sample Output:

f(x) = x * x

integral: 5.333333333344218


  • Easy to implement
  • Works well with functions that are symmetrical along Y axis


  • Requires great number of iterations to be accurate, especially for big intervals (huge value of b – a)
Java Implementation Matlab Implementation Python Implementation R Implementation C# Implementation C++ Implementation

Midpoint Rule (Rectangle Method) Algorithm

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