Numerical Integration (scipy.integrate)

Numerical integration calculates the area under a curve when an analytical anti-derivative does not exist or when working with discrete sensor streams.

1. Adaptive Quadrature: quad

scipy.integrate.quad wraps the Fortran QUADPACK library. It dynamically subdivides the integration domain using an adaptive Gauss-Kronrod 21-point rule, returning both the integral value and an upper bound on the absolute error:

result, abserr = integrate.quad(f, a, b)

Basic Integral, Infinite Limits & Discontinuities

2. Integrating Sampled Data: trapezoid & simpson

When integrating discrete data points $(x_i, y_i)$ from an experiment, you cannot sample arbitrary points:

from scipy.integrate import trapezoid
area = trapezoid(y, x=x)

from scipy.integrate import simpson
area = simpson(y, x=x)

Comparing Convergence on Sampled Sines

Let’s integrate $\int_0^\pi \sin(x) dx = 2.0$:

3. Double & Multiple Integrals (dblquad)

For 2D double integrals $\iint f(y, x) dy dx$, inner limits can depend dynamically on the outer variable:

4. Try It Yourself

Problem: Work Done by Isothermal Expansion

An ideal gas undergoes isothermal expansion from $V_1 = 1\text{ L}$ ($10^{-3}\text{ m}^3$) to $V_2 = 3\text{ L}$ ($3 \times 10^{-3}\text{ m}^3$) at $T = 300\text{ K}$ for $n = 1\text{ mol}$. Compute the expansion work $W = \int_{V_1}^{V_2} \frac{n R T}{V} dV$ using quad.

from scipy import integrate, constants
 
n, T = 1.0, 300.0
V1, V2 = 1.0e-3, 3.0e-3
 
W, err = integrate.quad(lambda V: n * constants.R * T / V, V1, V2)
print(f"Work W: {W:.2f} J")