Understanding the Trapezoidal Rule
The Trapezoidal Rule is a numerical integration method used to approximate the definite integral of a function. Instead of finding the exact area under a curve using anti-derivatives, we approximate the region by dividing the interval into $n$ smaller trapezoids.
The Formula
For an integral $\int_{a}^{b} f(x) dx$ with $n$ subintervals, the width of each trapezoid is $\Delta x = \frac{b-a}{n}$. The approximation formula is:
$$I \approx \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]$$
Solving the Problem: $\int_{0}^{4} x^3 dx$ with $n=4$
1. Identify parameters:
- $a = 0$, $b = 4$, $n = 4$
- $\Delta x = \frac{4-0}{4} = 1$
2. Find the points $x_i$:
- $x_0 = 0$
- $x_1 = 1$
- $x_2 = 2$
- $x_3 = 3$
- $x_4 = 4$
3. Calculate function values $f(x_i) = x_i^3$:
- $f(x_0) = 0^3 = 0$
- $f(x_1) = 1^3 = 1$
- $f(x_2) = 2^3 = 8$
- $f(x_3) = 3^3 = 27$
- $f(x_4) = 4^3 = 64$
4. Apply the Trapezoidal Rule:
$$I \approx \frac{1}{2} [f(0) + 2f(1) + 2f(2) + 2f(3) + f(4)]$$ $$I \approx \frac{1}{2} [0 + 2(1) + 2(8) + 2(27) + 64]$$ $$I \approx \frac{1}{2} [0 + 2 + 16 + 54 + 64] = \frac{1}{2} [136] = 68$$
The estimated value is $68$. For comparison, the exact integral is $[\frac{x^4}{4}]_0^4 = \frac{256}{4} = 64$.