Integration: The Art of Accumulation

While derivatives break a function down into its local rates of change, integration builds it back up. Integration is the process of adding up infinitely many tiny pieces to find a whole.

1. The Definite Integral as a Sum

We define the definite integral $\int_a^b f(x) \, dx$ as the signed area under the curve $f(x)$ from $a$ to $b$. Formally, this is reached through a Riemann Sum: we divide the area into $n$ rectangles and take the limit as $n \to \infty$.

Instead of just looking at the formula, let’s watch the approximation get better as we add more rectangles.

import numpy as np
import matplotlib.pyplot as plt

def f(x): return x * np.sin(x) + 2

a, b = 0, 10
n = 10 # Change this to 5, 20, or 100 to see convergence

x = np.linspace(a, b, 1000)
x_rect = np.linspace(a, b, n, endpoint=False)
width = (b - a) / n

plt.plot(x, f(x), 'k', linewidth=2)
plt.bar(x_rect, f(x_rect), width=width, align='edge', alpha=0.3, color='blue', edgecolor='b')
plt.title(f"Riemann Sum Approximation (n={n})")
plt.show()

2. Numerical Integration: Trapezoidal Rule

Computer scientists rarely integrate by hand. They use algorithms like the Trapezoidal Rule, which fits a line (trapezoid) between points instead of a flat rectangle. This usually converges much faster.

import numpy as np

def f(x): return np.exp(-x**2) # The Gaussian bell curve

a, b = 0, 1
n = 5
h = (b - a) / n

x = np.linspace(a, b, n + 1)
y = f(x)

# Trapezoidal rule: h/2 * (y0 + 2y1 + ... + yn)
area = (h/2) * (y[0] + 2 * np.sum(y[1:-1]) + y[-1])

print(f"Estimated area under e^(-x^2) from 0 to 1: {area:.6f}")
print("This is related to the Error Function (erf) used in Statistics.")

3. The Fundamental Theorem of Calculus (FTC)

The FTC states that integration and differentiation are inverse operations. If $F(x)$ is the antiderivative of $f(x)$, then: $\int_a^b f(x) \, dx = F(b) - F(a)$

This turns a hard problem of “infinite summing” into a simple problem of “subtraction.”

4. Improper Integrals

Sometimes we need to integrate over an infinite interval, such as $\int_0^\infty e^{-x} \, dx$. These are vital for calculating total energy or probabilities in bell curves.

5. Summary Check