Introduction to the Trapezoidal Rule
The Trapezoidal Rule is a technique for approximating the definite integral $\int_{a}^{b} f(x) dx$. Instead of finding the exact antiderivative, we approximate the area under the curve by dividing it into $n$ trapezoids of equal width.
The Formula
Given the interval $[a, b]$ divided into $n$ sub-intervals of width $h = \frac{b-a}{n}$, the Trapezoidal Rule is given by:
$$\int_{a}^{b} f(x) dx \approx \frac{h}{2} [y_0 + 2(y_1 + y_2 + ... + y_{n-1}) + y_n]$$
where $y_i = f(x_i)$ and $x_i = a + ih$.
Solving the Problem
We need to evaluate $\int_{0}^{2} \frac{dx}{1 + x^4}$ with $n=4$.
1. Identify parameters
- $a = 0, b = 2, n = 4$
- $h = \frac{b-a}{n} = \frac{2-0}{4} = 0.5$
2. Tabulate values
We evaluate $f(x) = \frac{1}{1+x^4}$ at $x_i = 0, 0.5, 1, 1.5, 2$:
| $i$ | $x_i$ | $y_i = f(x_i)$ |
|---|---|---|
| 0 | 0 | $1 / (1+0^4) = 1.000$ |
| 1 | 0.5 | $1 / (1+0.5^4) = 1 / 1.0625 \approx 0.9412$ |
| 2 | 1.0 | $1 / (1+1^4) = 1 / 2 = 0.500$ |
| 3 | 1.5 | $1 / (1+1.5^4) = 1 / 6.0625 \approx 0.1649$ |
| 4 | 2.0 | $1 / (1+2^4) = 1 / 17 \approx 0.0588$ |
3. Apply the rule
$$\text{Integral} \approx \frac{0.5}{2} [1.000 + 2(0.9412 + 0.500 + 0.1649) + 0.0588]$$ $$\text{Integral} \approx 0.25 [1.000 + 2(1.6061) + 0.0588]$$ $$\text{Integral} \approx 0.25 [1.000 + 3.2122 + 0.0588] = 0.25 [4.2710] = 1.06775$$
Rounding to 3 decimal places, we get 1.068.