Introduction to Simpson's 1/3 Rule
Numerical integration is essential when we cannot find an elementary antiderivative for a function. Simpson's 1/3 Rule is a powerful technique that approximates the area under a curve by fitting parabolic segments to data points.
The Formula
For an integral $\int_{a}^{b} f(x) \, dx$ divided into $n$ sub-intervals of width $h$, where $n$ must be an even number:
$$\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4(f(x_1) + f(x_3) + \dots + f(x_{n-1})) + 2(f(x_2) + f(x_4) + \dots + f(x_{n-2})) + f(x_n) \right]$$
Solving the Problem
We are given the integral $\int_{0}^{1} \sqrt{1+2x^2} \, dx$ with $h = 0.25$.
1. Identify intervals
The range is $[0, 1]$ with width $h=0.25$. Therefore, the number of intervals is $n = (1-0)/0.25 = 4$ (which is even, satisfying the requirement).
2. Tabulate values
We evaluate $f(x) = \sqrt{1+2x^2}$ at each step:
| $x$ | $f(x) = \sqrt{1+2x^2}$ |
|---|---|
| $x_0 = 0$ | $\sqrt{1+0} = 1.0000$ |
| $x_1 = 0.25$ | $\sqrt{1+2(0.0625)} \approx 1.0607$ |
| $x_2 = 0.50$ | $\sqrt{1+2(0.25)} \approx 1.2247$ |
| $x_3 = 0.75$ | $\sqrt{1+2(0.5625)} \approx 1.4577$ |
| $x_4 = 1.00$ | $\sqrt{1+2(1)} \approx 1.7321$ |
3. Apply the Rule
$$\text{Area} \approx \frac{0.25}{3} \left[ f(x_0) + 4(f(x_1) + f(x_3)) + 2(f(x_2)) + f(x_4) \right]$$
$$\text{Area} \approx \frac{0.25}{3} \left[ 1 + 4(1.0607 + 1.4577) + 2(1.2247) + 1.7321 \right]$$
$$\text{Area} \approx \frac{0.25}{3} \left[ 1 + 4(2.5184) + 2.4494 + 1.7321 \right]$$
$$\text{Area} \approx \frac{0.25}{3} \left[ 1 + 10.0736 + 2.4494 + 1.7321 \right] = \frac{0.25}{3} \times 15.2551 \approx 1.2713$$
Conclusion
The approximated value of the integral is approximately 1.2713.