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# Midpoint Rule Calculator

To use midpoint rule calculator, enter function, boundary points, subintervals, and click calculate button

Table of Contents:

## Midpoint Rule Calculator

Midpoint rule calculator is used to find the approximate definite integral over the given interval.

## What is the midpoint rule?

**Midpoint rule**, also known as the midpoint approximation. It is the numerical integration technique that approximates the area under the curve** f(x)** by dividing the interval into subintervals of equal length and evaluating the functional value at the midpoint of each subinterval.

**Formula**

The formula of the midpoint rule of the given function is:

_{a}∫^{b}f(x) dx= Δx/2[f(x_{0}+x_{1})/2+ f(x_{1}+x_{2})/2+…+ f(x_{n-2}+x_{n-1})/2+f(x_{n-1}+x_{n})/2]

where** **`Δx= (b-a)/n`

**f(x)**is the function whose midpoint is required.**a**and b are the lower limit and upper limit respectively.**Δx**is the length of the subinterval.**x**are the values of the subinterval._{0},x_{1,}x_{2}

**Example**

Find the midpoint of the function **2x ^{2 }- 3y** where the upper limit is

**3**, the lower limit is

**2**and the subinterval is

**3**.

**Solution:**

**Step 1: **The left Riemann sum rule for the endpoints:

_{a}∫^{b}f(x) dx= Δx/2[f(x_{0}+x_{1})/2 + f(x_{1}+x_{2})/2 +…+ f(x_{n-2}+x_{n-1})/2 + f(x_{n-1}+x_{n})/2]

We have that **f(x) = 2x ^{2}-3y,**

**a= 2**,

**b= 3**and

**n=3**

Therefore, `Δx= (3-2)/3=1/3`

**Step 2:** Find the subintervals

**[2,3]** interval is divided into **n=3** subintervals of the length **1/3**.

`[2,7/3,8/3,3]`

**Step 3: **Calculate the value

f(x_{0}+x_{1})/2= f(2+7/3)/2 = f(13/6) = (169/18)-3y

f(x_{1}+x_{2})/2= f(7/3+8/3)/2 = f(5/2) = (25/2)-3y

`f(x`

_{2}+x_{3})/2= f(8/3+3)/2 = f(17/6) = (289/18)-3y

**Step 4: **Now put all the values in the formula, we have

=1/3[(169/18)-3y+(25/2)-3y+(289/18)-3y]

`=12.65-3y`