The Simpson's 3/8 rule is a numerical integration method used to approximate the definite integral of a function. It's an extension of the Simpson's 1/3 rule, allowing for integration over a more extensive range of functions by approximating the curve with cubic approximations.
Basic Principle:
Similar to the Simpson's 1/3 rule, the 3/8 rule divides the interval into equally spaced segments. However, it fits cubic polynomials through every three consecutive segments instead of quadratic polynomials.
Steps in Simpson's 3/8 Rule:
1. Divide Interval into Segments:
- Divide the interval ([a, b]) into (n) equally spaced segments.
- Ensure (n) is a multiple of 3 to apply Simpson's 3/8 rule.
2. Calculate Approximation:
- Approximate the integral by summing the areas under cubic polynomials formed by every three consecutive segments.
- Calculate the area under each group of three segments using Simpson's 3/8 rule formula: ( text{Area} = frac{3h}{8} times [f(x_i) + 3f(x_{i+1}) + 3f(x_{i+2}) + f(x_{i+3})] ).
- Sum up these areas to obtain the approximate integral value.
Advantages and Limitations:
- Advantages:
- Higher accuracy: Offers higher accuracy for a broader range of functions compared to the Simpson's 1/3 rule.
- Efficiency: Requires fewer function evaluations than some other numerical integration techniques.
- Limitations:
- Requirement of a multiple of 3 segments: Simpson's 3/8 rule demands a multiple of 3 segments for application, limiting its use for cases with a different number of segments.
C++ Code for Simpson's 3/8 Rule:
```cpp
#include <iostream>
#include <cmath>
using namespace std;
Function to integrate (Example function: x^3)
double function(double x) {
return x * x * x;
}
Simpson's 3/8 rule for numerical integration
double simpsons3_8Rule(double a, double b, int n) {
double h = (b - a) / n;
double sum = function(a) + function(b);
for (int i = 1; i < n; i += 3) {
double x = a + i * h;
sum += 3 * function(x);
}
for (int i = 2; i < n - 1; i += 3) {
double x = a + i * h;
sum += 3 * function(x);
}
for (int i = 3; i < n - 2; i += 3) {
double x = a + i * h;
sum += 2 * function(x);
}
return (3 * h / 8) * sum;
}
int main() {
double a = 0.0; // Lower limit of integration
double b = 2.0; // Upper limit of integration
int n = 9; // Number of segments (should be a multiple of 3)
double integral = simpsons3_8Rule(a, b, n);
cout << "Approximate integral using Simpson's 3/8 Rule: " << integral << endl;
return 0;
}
Explanation:
- `function()` Function:
- Represents the function to be integrated. Modify this function to integrate a different function.
- `simpsons3_8Rule()` Function:
- Implements Simpson's 3/8 rule for numerical integration.
- Divides the interval ([a, b]) into (n) segments and calculates the approximate integral using the Simpson's 3/8 rule formula.
- `main()` Function:
- Sets the interval ([a, b]) and the number of segments (n) for the integration.
- Calls the `simpsons3_8Rule()` function to approximate the integral and displays the result.
Example:
The provided code demonstrates Simpson's 3/8 rule to approximate the integral of the function (f(x) = x^3) from 0 to 2 using 9 segments. It calculates and displays the approximate integral value obtained using Simpson's 3/8 rule. Adjust the function and interval as needed for different integrals.
Simpson's 3/8 rule is a more advanced numerical integration method, offering higher accuracy for a broader range of functions compared to simpler methods. Adjusting the number of segments can improve accuracy for more complex functions.
0 Comments