Question

Using Python

For example using Simpson 3/8 Rule:
def func(x):
return (float(1) / ( 1 + x * x ))
# Function to perform calculations
def calculate(lower_limit, upper_limit, interval_limit ):
interval_s

Answer 1

The Simpson's 3/8 rule is a method for numerical integration, similar to the simpler Simpson's 1/3 rule. It is a more accurate method and is used when higher precision is required. It involves dividing the area under a curve into multiple segments, calculating the area of each segment, and summing these areas to approximate the total integral.

The formula for Simpson's 3/8 rule is as follows:

`∫a to b f(x) dx ≈ (3h/8) * [f(a) + 3f(a+h) + 3f(a+2h) + f(b)]`

where 'h' is the width of the interval, 'a' is the lower limit, and 'b' is the upper limit.

Here's how you might implement this in Python, based on your provided function `func`:

```

def func(x):

return (float(1) / ( 1 + x * x ))

def calculate(lower_limit, upper_limit, interval_limit):

h = (upper_limit - lower_limit) / interval_limit

# calculation

integral = func(lower_limit) + func(upper_limit)

for i in range(1, interval_limit):

if i % 3 == 0:

integral += 2 * func(lower_limit + i * h)

else:

integral += 3 * func(lower_limit + i * h)

integral *= 3 * h / 8

return integral

```

In this code:

- `func` is the function you want to integrate.

- `calculate` performs the actual integration using Simpson's 3/8 rule.

- The integral is initialized with `func(lower_limit) + func(upper_limit)`.

- The loop then adds `2 * func(lower_limit + i * h)` if `i` is divisible by 3, and `3 * func(lower_limit + i * h)` otherwise.

- Finally, the integral is multiplied by `3 * h / 8` to get the final result.