5.3b Integration labs

Source: Lab ideas from Computational Physics (Landau, Páez, Bordeianu), §5.3–§5.3.5. Rewritten as original drills. Continues 5.3 Integration algorithms.

Build the $\sum f_i w_i$ skeleton: Riemann boxes, trapezoid / Simpson, Romberg, and a weight-sum audit. Stubs fail until filled in.

Formula refresh

Warm up: pick the matching equation among lookalikes (same walk as the theory lesson).

Labs

Activity: Left Riemann sum

Approximate $\int_a^b f(x)\,dx$ with $N$ equal boxes using left endpoints:

$h=(b-a)/N$, sum $= h\sum_{i=0}^{N-1} f(a+ih)$.

For $f(x)=e^{-x}$ on $[0,1]$, print the approximation with 10 decimals.

Input: N.

Write-ups

0/5 solved

5.3b.W1

Using “sum”, “weights”, and “nodes”, explain:

In one sentence: what do all numerical integration algorithms share?

Minimum 3 words. Your answer must include: “sum”, “weights”, “nodes”.

0 words/ 3 min

Meet the writing requirements, then you can open the model answer.

Explorers

# quad_ref.py
import math

def left_sum(f, a, b, N):
    h = (b - a) / N
    return h * sum(f(a + i * h) for i in range(N))

def mid_sum(f, a, b, N):
    h = (b - a) / N
    return h * sum(f(a + (i + 0.5) * h) for i in range(N))

def trap(f, a, b, N):
    h = (b - a) / N
    s = 0.5 * (f(a) + f(b))
    for i in range(1, N):
        s += f(a + i * h)
    return s * h

def simpson(f, a, b, N):
    h = (b - a) / N
    s = f(a) + f(b)
    for i in range(1, N, 2):
        s += 4 * f(a + i * h)
    for i in range(2, N, 2):
        s += 2 * f(a + i * h)
    return s * h / 3

def romberg(f, a, b, N):
    return (4 * trap(f, a, b, 2 * N) - trap(f, a, b, N)) / 3

Next: 5.4 Gaussian quadrature.