Numerical integration - Trapezoidal vs Simpson - Reversing the wheel of learning - High end computer science is plain Maths

1. The Core Idea

We want to approximate:

$$\int_a^b f(x)\,dx$$

by sampling the function at discrete points.

2. Trapezoidal Rule

Instead of rectangles, we use trapezoids.

${\int }_a^{b}f(x)dx\approx \frac{h}{2}[f(x_0)+2{\sum }_{i=1}^{n-1}f(x_i)+f(x_n)]$

Key properties:

• Error ~ O(h²)
• Assumes linear variation between points

👉 Think: “connect the dots with straight lines”

3. Simpson’s Rule

Now we approximate using parabolas (quadratic fit).

$\int_a^b f(x)\,dx \approx \frac{h}{3}\left[f(x_0) + 4\sum_{\text{odd}} f(x_i) + 2\sum_{\text{even}} f(x_i) + f(x_n)\right]$

Key properties:

• Error ~ O(h⁴) (much faster convergence)
• Requires even number of intervals

Think: “fit smooth curves instead of straight lines”

4. The Real Question: Which is Better?

Method	Accuracy	Cost	When it works best
Trapezoidal	Medium	Low	Rough/oscillatory data
Simpson	High	Medium	Smooth functions

5. A Perfect Test Function

Use something smooth but non-trivial:

$$f(x) = \sin(x), \quad \int_0^\pi \sin(x)\,dx = 2$$

7. Deep Insight (Important)

This experiment teaches:

• Numerical methods are about approximating shape
• Higher-order ≠ always better
• Smoothness of function matters
• There is a trade-off between truncation error and floating-point error