Understanding Simpson's 1/3 Rule
Simpson's 1/3 rule is a powerful method in numerical analysis used to approximate the definite integral of a function $f(x)$. It works by approximating the integrand with a quadratic polynomial over sub-intervals.
The Formula
For an integral $\int_{a}^{b} f(x) dx$ with $n$ sub-intervals of width $h = \frac{b-a}{n}$, the formula is:
$$\int_{a}^{b} f(x) dx \approx \frac{h}{3} [f(x_0) + 4f(x_1) + f(x_2)] \text{ (for } n=2\text{)}$$
Step-by-Step Solution
Given the integral $\int_{0}^{0.2} \sqrt{1-2x^2} dx$ with $n=2$:
1. Identify Parameters
- Interval $[a, b] = [0, 0.2]$
- Number of sub-intervals $n = 2$
- Width $h = \frac{0.2 - 0}{2} = 0.1$
2. Determine Data Points
We calculate $x_i$ and the corresponding $f(x_i) = \sqrt{1-2x_i^2}$:
- $x_0 = 0 \implies f(0) = \sqrt{1 - 2(0)^2} = 1$
- $x_1 = 0.1 \implies f(0.1) = \sqrt{1 - 2(0.1)^2} = \sqrt{1 - 0.02} = \sqrt{0.98} \approx 0.9899$
- $x_2 = 0.2 \implies f(0.2) = \sqrt{1 - 2(0.2)^2} = \sqrt{1 - 0.08} = \sqrt{0.92} \approx 0.9592$
3. Apply the Formula
$$\text{Integral} \approx \frac{0.1}{3} [f(0) + 4f(0.1) + f(0.2)]$$ $$\approx \frac{0.1}{3} [1 + 4(0.9899) + 0.9592]$$ $$\approx \frac{0.1}{3} [1 + 3.9596 + 0.9592] = \frac{0.1}{3} [5.9188]$$ $$\approx 0.19729$$
The approximate value of the integral is 0.1973.