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# Simpson's Rule Calculator

To use Simpson's Rule Calculator, select the method, put the values in the corresponding boxes, and hit the calculate button

## Simpson's Rule Calculator

Simpson's rule calculator is a helpful technique for evaluating the approximate values of the area under the curve with steps. This calculator follows the definite integral method as the upper and lower limits are involved in it.

## What is Simpson’s rule?

Simpson’s method is a numerical method to find the definite integral of a function in the given interval. The approximate area under the curve of the function by using a series of straight lines that connect the functions with discrete points.

There are two methods to solve the functional values by using Simpson’s rule.

- Simpson’s 1/3 rule
- Simpson’s 3/8 rule

Let’s discuss one by one both methods.

**The formula of Simpson’s 1/3 rule:**

Suppose a definite integral function

$$\int _a^b\:f\left(x\right)dx$$

Make intervals of [a, b] into subintervals of even numbers n.

$$\Delta x=\frac{\left(b-a\right)}{n}$$

$$\int _a^b\:f\left(x\right)dx=\frac{\Delta x}{3}\left[f\left(x_0\right)+4f\left(x_1\right)+2f\left(x_2\right)+4f\left(x_3\right)+...+4f\left(x_{n-1}\right)+f\left(x_n\right)\right]$$

- f(x) is the function whose definite integral find.
- b is the upper limit and a is the lower limit of the function.
- ∆x is the difference between the limit.

The number of subintervals is 3 that’s why it is called Simpson’s 1/3 rule.

**The formula of Simpson’s 3/8 rule:**

One more rule of Simpson’s is known as Simpson’s 2nd rule is given as:

$$\int _a^b\:\:f\left(x\right)dx=\frac{3\Delta \:x}{8}\left[f\left(x_0\right)+2\left(f\left(x_3\right)+f\left(x_6\right)+...+f\left(x_{n-3}\right)\right)+3\left(f\left(x_1\right)+f\left(x_2\right)+f\left(x_4\right)+f\left(x_5\right)+...+f\left(x_{n-2}\right)+f\left(x_{n-1}\right)\right)+f\left(x_n\right)\right]$$

The fraction 3/8 in the name of the rule refers to the weight given to the midpoint term in approximation.

## How do we calculate Simpson’s method?

We have to understand Simpson’s rule with the help of some numerical examples.

**Example 1:**

Solve the definite integral of a function 2x^{2}-3x where the upper limit is 3, the lower limit is 2 and the number of subintervals is 2 by using Simpson’s 1/3 rule.

**Solution:**

**Step 1: **First we have to find the length of the interval.

∆x = (3-2)/2 = 0.5

**Step 2: **Construct the subintervals of length 0.5.

[2,2.5,3]

**Step 3:** Now evaluate the function on these endpoints.

f(x_{0}) = f(2) = 2(2^{2}) - 3(2) =2

f(x_{1}) = f (2.5) =4(2(2.5)^{2}-3(2.5)) = 20

f(x_{2}) = f (3) = 2(3)^{2}- 3(3)=9

**Step 4: **Sum up the above all values and multiply by ∆x/3 = 0.167

= 0.167(2 + 20 + 9)

= 5.177

**Example 2:**

Solve the definite integral by using Simpson’s 3/8 of x^{2}-1 with lower limit 1, upper limit 2, and a number of intervals are 6.

**Solution:**

**Step 1:** We have to find the norm of the interval.

(2-1)/6=0.167

**Step 2: **Now the subintervals are

[1,1.17,1.34,1.51,1.68,1.85,2]

**Step 3:** Now evaluate the function at endpoints.

f(x_{0}) = f (1) = 1^{2}-1=0

f(x_{1}) = f (1.17) = 3(1.17^{2}-1) =1.11

f(x_{2}) = f (1.34) = 3(1.34^{2}-1) =2.39

f(x_{3}) = f (1.51) = 2(1.51^{2}-1) = 2.56

f(x_{4}) = f (1.68) = 3(1.68^{2}-1) = 5.47

f(x_{5}) = f (1.85) = 3(1.85^{2}-1) = 7.27

f(x_{6}) = f (2) = 2^{2}-1 = 3

**Step 4: **Put the values in the formula and evaluate

= 3(1/6) (0+1.11+2.39+2.56+5.47+7.27+3)

=1.38