Introduction to the Trapezoidal Rule
The Trapezoidal Rule is a technique for approximating the definite integral of a function. Instead of finding the exact antiderivative, we approximate the area under the curve by dividing the interval into smaller sub-intervals and forming trapezoids.
The formula for the Trapezoidal Rule is: $$\int_{a}^{b} f(x) dx \approx \frac{h}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]$$ where $h = \frac{b-a}{n}$ is the width of each sub-interval.
Step-by-Step Solution
Problem: Evaluate $\int_{0}^{\pi/2} \sqrt{\sin x} dx$ using the Trapezoidal Rule with $n = 2$.
1. Identify parameters
- Interval: $[a, b] = [0, \pi/2]$
- Number of segments: $n = 2$
- Width of sub-intervals: $h = \frac{\pi/2 - 0}{2} = \frac{\pi}{4}$
2. Determine sample points
- $x_0 = 0$
- $x_1 = 0 + \pi/4 = \pi/4$
- $x_2 = \pi/4 + \pi/4 = \pi/2$
3. Evaluate the function $f(x) = \sqrt{\sin x}$ at sample points
- $f(x_0) = \sqrt{\sin(0)} = 0$
- $f(x_1) = \sqrt{\sin(\pi/4)} = \sqrt{1/\sqrt{2}} \approx \sqrt{0.7071} \approx 0.8409$
- $f(x_2) = \sqrt{\sin(\pi/2)} = \sqrt{1} = 1$
4. Apply the Trapezoidal Formula
$$\text{Area} \approx \frac{h}{2} [f(x_0) + 2f(x_1) + f(x_2)]$$ $$\text{Area} \approx \frac{\pi/4}{2} [0 + 2(0.8409) + 1]$$ $$\text{Area} \approx \frac{\pi}{8} [1.6818 + 1] = \frac{\pi}{8} (2.6818)$$ $$\text{Area} \approx 0.3927 \times 2.6818 \approx 1.0531$$
Conclusion
By using $n=2$ segments, we approximated the integral to be approximately $1.0531$. Increasing $n$ would lead to a more accurate result closer to the true value.