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Simpson's 1/3 rule

Simpson's 1/3 Rule Explained The Simpson's 1/3 rule is a technique for approximating the definite integral of a function by dividing the interval of integrat...

Simpson's 1/3 Rule Explained#

The Simpson's 1/3 rule is a technique for approximating the definite integral of a function by dividing the interval of integration into equal-width subintervals and using weighted averages of function values within each subinterval.

How it works:

Imagine dividing the interval $[a, b]$ into 3 equal-width subintervals of length $h = (b - a) / 3$ . This means the subintervals are:

$a = x_0, x_1 = a + h, x_2 = a + 2h, x_3 = b$

For each subinterval, we evaluate the function value $f(x)$ at the endpoints and midpoints of the interval.

The function value at the left endpoint is denoted by $f(x_0)$ .
The function value at the right endpoint is denoted by $f(x_3)$ .
The function value at the midpoint is denoted by $f(x_1)$ .

We then use the weighted average of these three function values to approximate the integral:

$\int_a^b f(x) d x \approx \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2)]$

Important points:

The weights for the weighted average are chosen to be uniformly spaced, meaning the distance between weights is equal. This leads to the rule being more accurate than the Trapezoidal rule in some cases.
Simpson's rule is more accurate than the Trapezoidal rule when the function is smooth and continuous on the interval.
It's important to choose the right number of subintervals (3 is typically enough) for good accuracy with the Simpson's rule.

Example:

Imagine we want to approximate the definite integral of the function $f(x) = x^2$ from $a = 0$ to $b = 1$ . Using 3 subintervals with equal width, we get:

$x_0 = 0, x_1 = 0.33, x_2 = 0.66, x_3 = 1$

and the corresponding function values are:

$f(x_0) = 0, f(x_1) = 0.09, f(x_2) = 0.36, f(x_3) = 1$

Plugging these values into the formula, we get:

$\int_0^1 x^2 d x \approx 0.55$

This is the approximate integral of the function using Simpson's 1/3 rule

Simpson's 1/3 Rule Explained#

How it works:

Imagine dividing the interval

[a, b]

into 3 equal-width subintervals of length

h = (b - a) / 3

. This means the subintervals are:

a = x_0, x_1 = a + h, x_2 = a + 2h, x_3 = b

For each subinterval, we evaluate the function value

f(x)

at the endpoints and midpoints of the interval.

The function value at the left endpoint is denoted by $f(x_0)$ .

The function value at the right endpoint is denoted by $f(x_3)$ .

The function value at the midpoint is denoted by $f(x_1)$ .

We then use the weighted average of these three function values to approximate the integral:

\int_a^b f(x) d x \approx \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2)]

Important points:

The weights for the weighted average are chosen to be uniformly spaced, meaning the distance between weights is equal. This leads to the rule being more accurate than the Trapezoidal rule in some cases.

Simpson's rule is more accurate than the Trapezoidal rule when the function is smooth and continuous on the interval.

It's important to choose the right number of subintervals (3 is typically enough) for good accuracy with the Simpson's rule.

Example:

Imagine we want to approximate the definite integral of the function

f(x) = x^2

from

a = 0

b = 1

. Using 3 subintervals with equal width, we get:

x_0 = 0, x_1 = 0.33, x_2 = 0.66, x_3 = 1

and the corresponding function values are:

f(x_0) = 0, f(x_1) = 0.09, f(x_2) = 0.36, f(x_3) = 1

Plugging these values into the formula, we get:

\int_0^1 x^2 d x \approx 0.55

This is the approximate integral of the function using Simpson's 1/3 rule

Simpson's 1/3 rule

Simpson's 1/3 Rule Explained#

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Simpson's 1/3 Rule Explained#