Topic: Python SciPy – From Easy to Top: Part 2 of 6: Numerical Integration and Differentiation

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1. Numerical Integration Overview

• Numerical integration approximates the area under curves when an exact solution is difficult or impossible.

• SciPy provides several methods like quad, dblquad, and trapz.

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2. Using `scipy.integrate.quad`

This function computes the definite integral of a function of one variable.

Example: Integrate cos(x) from 0 to pi divided by 2

import numpy as np
from scipy import integrate

result, error = integrate.quad(np.cos, 0, np.pi/2)
print("Integral of cos(x) from 0 to pi/2:", result)

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3. Double Integration with `dblquad`

Integrate a function of two variables over a rectangular region.

Example: Integrate f(x, y) = x times y over x from 0 to 1, y from 0 to 2

def f(x, y):
    return x * y

result, error = integrate.dblquad(f, 0, 1, lambda x: 0, lambda x: 2)
print("Double integral result:", result)

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4. Using the Trapezoidal Rule: `trapz`

Useful for integrating discrete data points.

Example:

import numpy as np
from scipy import integrate

x = np.linspace(0, np.pi, 100)
y = np.sin(x)

area = integrate.trapz(y, x)
print("Approximate integral using trapz:", area)

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5. Numerical Differentiation with `derivative`

SciPy’s derivative function approximates the derivative of a function at a point.

Example: Derivative of sin(x) at x equals pi divided by 4

from scipy.misc import derivative
import numpy as np

def f(x):
    return np.sin(x)

dx = derivative(f, np.pi/4, dx=1e-6)
print("Derivative of sin(x) at pi/4:", dx)

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6. Limitations of `derivative`

• derivative uses finite difference methods, which can be noisy for non-smooth functions.

• Suitable for simple derivative calculations but not for complex cases.

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7. Summary

• quad is powerful for one-dimensional definite integrals.

• dblquad handles two-variable integration.

• trapz approximates integration from sampled data.

• derivative provides numerical differentiation.

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Exercise

• Compute the integral of e to the power of negative x squared from 0 to 1 using quad.

• Calculate the derivative of cos(x) at 0.

• Use trapz to approximate the integral of x squared over \[0, 5] using 50 points.

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