Introduction to Simpson's Rule
Numerical integration is essential when a function is difficult or impossible to integrate analytically. Simpson's Rule approximates a definite integral by using quadratic polynomials to fit the curve of the function.
The Formula
For an integral $\int_{a}^{b} f(x) dx$ with an even number of sub-intervals $n$, the step size $h$ is given by: $h = \frac{b-a}{n}$
Simpson's 1/3 Rule formula is: $I \approx \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)]$
Step-by-Step Solution
We aim to evaluate $\int_{0}^{2} \frac{1}{1+x^4} dx$ with $n=4$.
1. Identify Parameters
- $a = 0$, $b = 2$, $n = 4$
- $h = \frac{2-0}{4} = 0.5$
2. Determine Data Points
- $x_0 = 0, f(0) = 1/(1+0) = 1$
- $x_1 = 0.5, f(0.5) = 1/(1+0.0625) = 1/1.0625 \approx 0.94118$
- $x_2 = 1.0, f(1.0) = 1/(1+1) = 0.5$
- $x_3 = 1.5, f(1.5) = 1/(1+5.0625) = 1/6.0625 \approx 0.16495$
- $x_4 = 2.0, f(2.0) = 1/(1+16) = 1/17 \approx 0.05882$
3. Calculate the Integral
$I \approx \frac{0.5}{3} [1 + 4(0.94118) + 2(0.5) + 4(0.16495) + 0.05882]$ $I \approx 0.16667 [1 + 3.76472 + 1 + 0.6598 + 0.05882]$ $I \approx 0.16667 [6.48334]$ $I \approx 1.08055$
Rounding to three decimal places, the result is 1.081.