Numerical Integration

🔹 Riemann Sum

Approximate area under $f(x)$ using rectangles.

General form:

$$\int_a^b f(x)\,dx \approx f(x_1^*)\Delta x + f(x_2^*)\Delta x + \cdots + f(x_n^*)\Delta x$$

• $\Delta x = \frac{b-a}{n}$
• $x_i^*$ can be:
  • Left: $x_1 = a$, $x_2 = a+\Delta x$, ..., $x_n = a+(n-1)\Delta x$
  • Right: $x_1 = a+\Delta x$, $x_2 = a+2\Delta x$, ..., $x_n = b$
  • Midpoint: $x_1 = a+\tfrac{1}{2}\Delta x$, $x_2 = a+\tfrac{3}{2}\Delta x$, ..., $x_n = b-\tfrac{1}{2}\Delta x$

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🔹 Trapezoidal Rule

Approximate area using trapezoids.

Formula:

$$\int_a^b f(x)\,dx \approx \frac{\Delta x}{2} \big[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \big]$$

• Points: $x_0 = a$, $x_1 = a+\Delta x$, ..., $x_n = b$
• More accurate than basic Riemann sums
• Uses linear approximation

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🔹 Simpson’s Rule

Uses parabolas (quadratics) for better accuracy.

Formula (n must be even):

$$\int_a^b f(x)\,dx \approx \frac{\Delta x}{3} \big[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \big]$$

• Points: $x_0 = a$, $x_1 = a+\Delta x$, ..., $x_n = b$
• Pattern: $1,4,2,4,2,\dots,4,1$
• Very accurate for smooth functions

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🔹 Quick Comparison

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🔹 Pro Tip

• If function is linear → trapezoidal is exact
• If function is quadratic → Simpson is exact
• Increasing $n$ → improves all methods